Lemma

Algebra · Chapter 14 of 20

Sequences & series

165 curated · 622 from open sources · 14.3 million generated

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Sum of an arithmetic seriesMedium · 14.3 million variationsTerm of a geometric sequenceMedium · 16,102 variations

Curated

165
#TitleDifficultySourceTags
12Integer echoesRecurrencesMediumLemmaInductionRecurrence+1
45A third at a timeSumsMediumLemmaGeometric series
813Arithmetic means between fractionsSequences & seriesMediumHall & Knightarithmetic progressionmeans
814Arithmetic means between mixed numbersSequences & seriesMediumHall & Knightarithmetic progressionmeans
815Arithmetic means with variable termSequences & seriesMediumHall & Knightarithmetic progressionmeans
821Three numbers in APSequences & seriesMediumHall & Knightarithmetic progressionproduct
822Numbers from sum of cubesSequences & seriesMediumHall & Knightarithmetic progressioncubes
826Sum of mixed quadratic seriesSequences & seriesMediumHall & Knightarithmetic progressionsummation
830First term and lengthSequences & seriesMediumHall & Knightarithmetic progressionterm formula
831First term and length from later termsSequences & seriesMediumHall & Knightarithmetic progressionterm formula
832First installment of a debt repaymentSequences & seriesMediumHall & Knightarithmetic progressionsum of series
833Number of arithmetic meansSequences & seriesMediumHall & Knightarithmetic progressionmeans
835Sum of mixed seriesSequences & seriesMediumHall & Knightseriessummation
836Sum of n termsSequences & seriesMediumHall & Knightarithmetic progressionsum formula
837Relation among three termsSequences & seriesMediumHall & Knightarithmetic progressionterm relation
838Sum of p+q termsSequences & seriesMediumHall & Knightarithmetic progressionsum
839Four integers in A.P.Sequences & seriesMediumHall & Knightarithmetic progressionproduct+1
840Divide 20 into A.P.Sequences & seriesMediumHall & Knightarithmetic progressionratio
841Term swapping in A.P.Sequences & seriesMediumHall & Knightarithmetic progressionterm formula
843General term from sum formulaSequences & seriesMediumHall & Knightsum formulageneral term
846Number of terms from partial sumsSequences & seriesMediumHall & Knighteven number of termspartial sums
847Relation for equal meansSequences & seriesMediumHall & Knightarithmetic meanrelation
952Third ProportionalSequences & seriesMediumHall & KnightProportionAlgebraic Fractions
953Proportion Identity Proof OneSequences & seriesMediumHall & KnightProportionAlgebraic Identities
954Proportion Identity Proof TwoSequences & seriesMediumHall & KnightProportionAlgebraic Identities
955Proportion Identity Proof ThreeSequences & seriesMediumHall & KnightProportionRadicals
968Four proportionals with given sumsSequences & seriesMediumHall & Knightproportionsystems of equations
969Mixing sherry from two casksSequences & seriesMediumHall & Knightratiolinear equations
970Capacity of a wine cask after two dilutionsSequences & seriesMediumHall & Knightproportiondilution
972Town versus country population in 1871Sequences & seriesMediumHall & Knightpercentagesratio
973Comparing percentage increases of tea and coffeeSequences & seriesMediumHall & Knightpercentageslinear equations
993Diamond and Gold Ring ValuesSequences & seriesMediumHall & KnightVariationProportionality
1004Sum of alternating geometric seriesSequences & seriesMediumHall & Knightgeometric progressionrationalizing denominators
1006Insert geometric meansSequences & seriesMediumHall & Knightgeometric meansgeometric progression
1007Inserting Geometric MeansSequences & seriesMediumHall & KnightGeometric progressionGeometric means
1013Sum of a geometric seriesSequences & seriesMediumHall & KnightGeometric progressionInfinite sum
1014Common ratio from partial sumsSequences & seriesMediumHall & KnightGeometric progressionPartial sums
1015Determine the GP from two termsSequences & seriesMediumHall & KnightGeometric progressionTerm formula
1016First term from sum and last termSequences & seriesMediumHall & KnightGeometric progressionSum formula
1017Common ratio from first, last and sumSequences & seriesMediumHall & KnightGeometric progressionSum of GP
1018Three numbers in GP from sum and productSequences & seriesMediumHall & KnightGeometric progressionSum and product
1019Three numbers in geometric progressionSequences & seriesMediumHall & KnightGeometric progressionSymmetric sums
1020Relation between geometric series sumsSequences & seriesMediumHall & KnightGeometric seriesSeries manipulation
1021Exponential relation of GP termsSequences & seriesMediumHall & KnightGeometric progressionExponents
1029Compare terms of two progressionsSequences & seriesMediumHall & Knightgeometric progressionproof
1030Sum of geometric series equalitySequences & seriesMediumHall & Knightgeometric progressionseries sum+1
1031Sum of complicated infinite seriesSequences & seriesMediumHall & Knightinfinite seriesgeometric series
1032Three numbers in geometric progressionSequences & seriesMediumHall & Knightgeometric progressionarithmetic progression+1
1033Sum of a combined seriesSequences & seriesMediumHall & Knightgeometric progressionarithmetic progression+1
1034Sum of mixed geometric termsSequences & seriesMediumHall & KnightGeometric progressionSeries summation
1035Alternating arithmetic‑geometric seriesSequences & seriesMediumHall & KnightArithmetic progressionGeometric progression
1038Relation among four consecutive G.P. termsSequences & seriesMediumHall & KnightGeometric progressionAlgebraic identity
1039Arithmetic and geometric mean conditionSequences & seriesMediumHall & KnightArithmetic meanGeometric mean
1040Sum of arithmetico-geometric seriesSequences & seriesMediumHall & Knightseriesgeometric progression+1
1041Alternating ratio seriesSequences & seriesMediumHall & Knightgeometric progressionsummation+1
1042Sum of partial sums of GPSequences & seriesMediumHall & Knightgeometric progressionsum of sums
1043Sum of infinite geometric seriesSequences & seriesMediumHall & Knightinfinite geometric seriessummation
1047Numbers with given meansSequences & seriesMediumHall & KnightGeometric meanHarmonic mean
1049Proportion in harmonic progressionSequences & seriesMediumHall & KnightHarmonic progressionProportion
1050Harmonical Progression PropertySequences & seriesMediumHall & Knightharmonic progressionarithmetic progression
1051Terms of a Harmonical ProgressionSequences & seriesMediumHall & Knightharmonic progressionalgebraic identity
1056Sum of a cubic‑quadratic seriesSequences & seriesMediumHall & KnightSummationPolynomial series
1057Sum of exponential differenceSequences & seriesMediumHall & KnightGeometric seriesSummation
1061Nth term from quadratic sumSequences & seriesMediumHall & KnightSummationArithmetic progression
1063Means ratio identitySequences & seriesMediumHall & Knightmeansproportionality
1067Rectangular pyramidal cannonballsSequences & seriesMediumHall & Knightrectangular pyramidal numbers
1068Incomplete triangular pyramidSequences & seriesMediumHall & Knightpartial tetrahedral numbers
1069Incomplete square pyramid with given coursesSequences & seriesMediumHall & Knightpartial square pyramidal numbers
1070Find length from rectangular pyramid totalSequences & seriesMediumHall & Knightrectangular pyramidal numberssolving for dimension
1071Shot count in square pileSequences & seriesMediumHall & Knightharmonic progressionseries
1072Complete rectangular pileSequences & seriesMediumHall & Knightharmonic progressionrectangular pile
1073Incomplete rectangular pileSequences & seriesMediumHall & Knightharmonic progressionrectangular pile
1074Complete rectangular pile from topSequences & seriesMediumHall & Knightharmonic progressionrectangular pile
1075Triangular vs square pileSequences & seriesMediumHall & Knightharmonic progressiontriangular pile
1076Incomplete square pileSequences & seriesMediumHall & Knightharmonic progressionsquare pile
1077Shot count in square vs triangular pileSequences & seriesMediumHall & Knighttriangular numberssquare numbers
1078Shot numbers from given ratioSequences & seriesMediumHall & Knightratiotriangular numbers+1
1079Lowest layer of a triangular shot pileSequences & seriesMediumHall & Knightunit pricetriangular numbers
1080Remaining shot forms a triangular pileSequences & seriesMediumHall & Knightsquare numberstriangular numbers
1387Binomial Expansion with Negative PowersSequences & seriesMediumHall & KnightBinomial TheoremNegative Exponents
1397Fifth term with fractional powersSequences & seriesMediumHall & Knightbinomial theoremfractional exponents
1399Difference of fifth powersSequences & seriesMediumHall & Knightbinomial theoremsymmetry
1400Difference of sixth powers with radicalsSequences & seriesMediumHall & Knightbinomial theoremsurds
1401Sum of sixth powers with a variableSequences & seriesMediumHall & Knightbinomial theorempolynomial expansion
1408Coefficient of x to the rSequences & seriesMediumHall & Knightbinomial theoremcombinatorial coefficients
1409Term Independent for MultiplesSequences & seriesMediumHall & Knightbinomial theoremindependent term
1410Coefficient of x to the pSequences & seriesMediumHall & Knightbinomial theoremproof
1423Term from the EndSequences & seriesMediumHall & KnightBinomial TheoremBinomial Coefficients
1425Relation Between IndicesSequences & seriesMediumHall & KnightBinomial TheoremCombinations
1426Middle Term of BinomialSequences & seriesMediumHall & KnightBinomial TheoremFactorials
1428Sum of Weighted Coefficients IISequences & seriesMediumHall & KnightBinomial TheoremIntegration
1443Binomial expansion with leading constantSequences & seriesMediumHall & KnightBinomial TheoremSeries Expansion
1444Binomial expansion with square root of a constantSequences & seriesMediumHall & KnightBinomial TheoremSeries Expansion
1446Binomial Expansion with Negative CoefficientSequences & seriesMediumHall & Knightbinomial theoremseries expansion
1449Specific Term with Higher PowerSequences & seriesMediumHall & Knightbinomial theoremgeneral term
1450Specific Term of Rational PowerSequences & seriesMediumHall & Knightbinomial theoremgeneral term
1453Specific term of scaled binomial powerSequences & seriesMediumHall & Knightbinomial theoremscaling
1454Seventh term of binomial expansionSequences & seriesMediumHall & KnightBinomial TheoremGeneral Term
1459Greatest term in binomial expansionSequences & seriesMediumHall & KnightBinomial theoremGreatest term
1460Greatest term in fractional binomial expansionSequences & seriesMediumHall & KnightBinomial theoremGreatest term
1461Greatest term in negative fractional binomialSequences & seriesMediumHall & KnightBinomial theoremGreatest term
1462Greatest term with two variablesSequences & seriesMediumHall & KnightBinomial theoremGreatest term
1463Two equal greatest terms in binomial expansionSequences & seriesMediumHall & KnightBinomial theoremGreatest term
1468Fourth root approximationSequences & seriesMediumHall & KnightBinomial theoremApproximation
1469Reciprocal cube rootSequences & seriesMediumHall & KnightBinomial theoremApproximation
1470Fractional power approximationSequences & seriesMediumHall & KnightBinomial theoremApproximation
1477Approximation with Small QuantitiesSequences & seriesMediumHall & KnightBinomial TheoremApproximation
1478Fractional ApproximationSequences & seriesMediumHall & KnightBinomial TheoremApproximation
1479Coefficient in Negative Power ExpansionSequences & seriesMediumHall & KnightBinomial TheoremCombinatorics
1481First Three Terms of ProductSequences & seriesMediumHall & KnightBinomial TheoremPower Series
1482First Three Terms of Rational CombinationSequences & seriesMediumHall & KnightBinomial TheoremPower Series
1483Binomial coefficient relationSequences & seriesMediumHall & Knightbinomial theoremseries expansion
1486General coefficient of rational fractionSequences & seriesMediumHall & Knightpartial fractionsbinomial theorem
1487General coefficient with cubic factorSequences & seriesMediumHall & Knightbinomial theorempower series
1488Prove an infinite series identitySequences & seriesMediumHall & Knightbinomial theoremseries summation
1489Prove a root series identitySequences & seriesMediumHall & Knightbinomial theoremseries summation
1490Binomial Series Identity TransformationSequences & seriesMediumHall & KnightBinomial TheoremSeries Identities
1491Equivalence of Binomial SeriesSequences & seriesMediumHall & KnightBinomial TheoremSeries Identity
1492Approximate Expansion of FunctionSequences & seriesMediumHall & KnightBinomial TheoremApproximations
1493Parity of Surd Power Integer PartSequences & seriesMediumHall & KnightBinomial TheoremDivisibility
1494Parity of Another Surd PowerSequences & seriesMediumHall & KnightBinomial TheoremDivisibility
1495Coefficient in Reciprocal SeriesSequences & seriesMediumHall & KnightBinomial TheoremGenerating Functions
1500Sum of Products of Binomial CoefficientsSequences & seriesMediumHall & KnightBinomial CoefficientsCombinatorial Identities
1502Alternating Harmonic Series of Binomial CoefficientsSequences & seriesMediumHall & KnightBinomial CoefficientsIntegration
1534Sum of Natural Numbers Series ExpansionSequences & seriesMediumHall & KnightBinomial TheoremGenerating Functions
1550Coefficient in multinomial expansionSequences & seriesMediumHall & KnightMultinomial TheoremPower Series
1551Coefficient of high powerSequences & seriesMediumHall & KnightMultinomial TheoremBinomial Series
1552Coefficient from rational expressionSequences & seriesMediumHall & KnightMultinomial TheoremGenerating Functions
1554Power series expansion up to quintic termSequences & seriesMediumHall & KnightBinomial TheoremPower Series
1555Series expansion with constant multiplierSequences & seriesMediumHall & KnightBinomial TheoremPower Series
1630Amount of an AnnuitySequences & seriesMediumHall & Knightcompound interestannuity+1
1634Amount of Annuity for Two YearsSequences & seriesMediumHall & Knightperpetual annuitycompound interest+1
1635Perpetual and Term AnnuitiesSequences & seriesMediumHall & Knightannuitiescompound interest+1
1636Deferred Freehold EstateSequences & seriesMediumHall & Knightannuitiespresent value+1
1639Repayment by InstalmentsSequences & seriesMediumHall & Knightloansannuities+1
1641Renewing a LeaseSequences & seriesMediumHall & Knightcompound interestannuities+1
1643Perpetual Increasing AnnuitySequences & seriesMediumHall & Knightarithmetico-geometric seriesperpetual annuity
1717Sum of odd squaresSequences & seriesMediumHall & Knightseries sumundetermined coefficients+1
1718Sum of consecutive productsSequences & seriesMediumHall & Knightseries sumundetermined coefficients
1719Sum of mixed productsSequences & seriesMediumHall & Knightseries sumundetermined coefficients
1720Sum of odd cubesSequences & seriesMediumHall & Knightseries sumundetermined coefficients
1721Sum of fourth powersSequences & seriesMediumHall & Knightseries sumundetermined coefficients
1732Determine Parameters for a General TermSequences & seriesMediumHall & Knightbinomial theoremundetermined coefficients+1
1733Coefficients in Rational ExpansionSequences & seriesMediumHall & KnightUndetermined coefficientsGenerating functions+1
1734Series Expansion of Implicit FunctionSequences & seriesMediumHall & KnightUndetermined coefficientsPower series
1748General Term of ExpandoSequences & seriesMediumHall & Knightpartial fractionsbinomial theorem+1
1749General Term of Partial FractionsSequences & seriesMediumHall & KnightPartial fractionsBinomial expansion+1
1750General Term of Rational FunctionSequences & seriesMediumHall & KnightPartial fractionsBinomial expansion+1
1753General term with repeated factorSequences & seriesMediumHall & KnightPartial fractionsBinomial theorem
1754General term of three linear factorsSequences & seriesMediumHall & KnightPartial fractionsGeometric series
1756Sum of infinite alternating power seriesSequences & seriesMediumHall & KnightTelescoping seriesInfinite series
1759Generating function with alternating signsSequences & seriesMediumHall & Knightrecurring seriesgenerating function
1760Generating function of an exponential-like seriesSequences & seriesMediumHall & Knightrecurring seriesgenerating function
1763Sum of a recurring seriesSequences & seriesMediumHall & Knightrecurring seriessummation
1764Recurring series term and sumSequences & seriesMediumHall & Knightrecurring seriesgenerating function+1
1766Alternate recurring seriesSequences & seriesMediumHall & Knightrecurring seriessum to n terms
1767Scales of relation for powersSequences & seriesMediumHall & Knightrecurring seriesfinite differences
1768Sum of recurring series from infinitySequences & seriesMediumHall & Knightrecurring seriesinfinite sums
1769Sum of a complex seriesSequences & seriesMediumHall & Knightrecurring seriessummation+1
1887Finite Series Identity OneSequences & seriesMediumHall & KnightFinite differencesBinomial theorem
1888Finite Series Identity TwoSequences & seriesMediumHall & KnightFinite differencesBinomial theorem
1889Finite Series Identity ThreeSequences & seriesMediumHall & KnightFinite differencesFactorials
1890Finite Series Identity FourSequences & seriesMediumHall & KnightFinite differencesBinomial theorem
1891Sum of series with telescoping terms ISequences & seriesMediumHall & KnightTelescoping seriesPartial fractions
1905Deciphering the ManuscriptSequences & seriesMediumHall & Knightarithmetic progressionword problems+1

From open sources

622

Showing everything, including problems with no answer to check, a missing figure, or a near-copy elsewhere. Show only complete problems

#TitleDifficultySource
11393Cody was reading through his favorite book seriesMediumGSM8K
12407The space station, Lupus-1, is an enormous spacecraft made up of three…MediumGSM8K
15553Jared wants to watch a series with four episodesMediumGSM8K
16742While chatting in the school cafeteria, Cesar's friend Demi tells him that…MediumGSM8K
19029Find the number that fits somewhere into the middle of the seriesMediumAQuA
19248What is the next number in this seriesMediumAQuA
19569Look at this series: V, VIII, XI, XIV, XVII, __, MediumAQuA
19680Find the term next in series MediumAQuA
19773The 52nd term of the series a, b , b, c, c…MediumAQuA
19871Find the number that fits somewhere into the middle of the seriesMediumAQuA
19992find the next term in the series 1, 0, -2, -3, 8…MediumAQuA
20388For positive integers n, the integer part of the nth term of…MediumAQuA
20605Look at this series: V, __, XI, XIV, XVII, XX, MediumAQuA
20957Can you replace the question mark with the number to complete the…MediumAQuA
21119The nth term in a certain sequence is defined for positive integer…MediumAQuA
21327n a series of numbers , the next number is formed by…MediumAQuA
21461Can you complete below number series by replacing "?" with the correct…MediumAQuA
22041Consider the sequence of numbers beginning 23, 27, 28, 32, 33MediumAQuA
22484Look at this series: U32, V29, __, X23, Y20, MediumAQuA
22862Find the term next in seriesMediumAQuA
22882What number comes next in this Number seriesMediumAQuA
23205What number comes next in this Number seriesMediumAQuA
23624Look at this series: V, VIII, XI, XIV, __, XX, MediumAQuA
24256Series T is a sequence of numbers where each term after the…MediumAQuA
24500In each series, look for the degree and direction of change between…MediumAQuA
24783In each series, look for the degree and direction of change between…MediumAQuA
24941Look at this series: 7, 10, 8, 11, 9, 12, 10, 13…MediumAQuA
25028What will come in place of the x in the following Number…MediumAQuA
25466What will come in place of the x in the following Number…MediumAQuA
25556Sum of 3 consecutive even no.'s is 26 more than the 1st…MediumAQuA
25594In a series of six consecutive odd numbers, the sum of the…MediumAQuA
25712Find 16th term in the series 7, 13, 19, 25, MediumAQuA
25989What is the next number in the given series 4,18,100,294MediumAQuA
26785What will come in place of the x in the following Number…MediumAQuA
2722187th number in the series 2, 10, 26,50MediumAQuA
27236What will come in place of the x in the following Number…MediumAQuA
27321Find the number that fits somewhere into the middle of the seriesMediumAQuA
27370Find the invalid no.from the following series 2,3,12,37,86,166,288MediumAQuA
27553In a sequence of 12 numbers, each term, except for the first…MediumAQuA
277371 2 1 2 2 1 2 2 2 1 2 2…MediumAQuA
27845Find out the wrong term in the series 1, 10, 100 100…MediumAQuA
28212Which number replace question mark in the seriesMediumAQuA
28393Look at this series: 15, 15, 27, 27, 39, 39, 51, 51…MediumAQuA
28410What are the last three number of this series 11234783131MediumAQuA
28626Look at this series: 80, 10, 70, 15, 60, MediumAQuA
29153Look at this series: 3, 1, (1/3), (1/9)MediumAQuA
30234What will come in place of the x in the following Number…MediumAQuA
30273Whats the next number in this number series MediumAQuA
304180, 0, 1, 2, 2, 4, 3, 6, 4, _ Can you…MediumAQuA
30628Consider a sequence of numbers given by the expression 7 + (h…MediumAQuA
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