Algebra

Problems related to equations, functions, polynomials, etc.

Problems in this category

The problem involves a polynomial $F(x, y, z) = x^3 + y^3 + z^3 - 3xyz$. (a) We need to show that $F...

PolynomialsFactorizationCyclic ExpressionsAlgebraic Manipulation

We are given an arithmetic sequence $\{a_n\}$ with the sum of the first $n$ terms denoted by $S_n$. ...

Arithmetic SequencesSeriesSummationLinear Equations

The problem is to find the first term $a_1$ and the common difference $d$ of an arithmetic sequence ...

Arithmetic SequencesSeriesLinear Equations

Given an arithmetic sequence $\{a_n\}$, we know that $a_2 + a_8 = 1$. The sum of the first 3 terms i...

Arithmetic SequenceSeriesSummation

The problem states that ${a_n}$ is an arithmetic sequence. We are given that $a_1 = \frac{1}{2}$ and...

Arithmetic SequencesSeriesCommon DifferenceSummation

Problem 17: Given an arithmetic sequence $\{a_n\}$ with $a_5 = 1$ and $a_8 = 8$, find the common dif...

SequencesArithmetic SequencesGeometric SequencesSeriesCommon DifferenceSum of Terms

We are given a geometric sequence $\{a_n\}$ where $a_1 = 3$. We also know that $4a_1, 2a_2, a_3$ for...

Sequences and SeriesGeometric SequenceArithmetic SequenceCommon RatioSolving Equations

Let $S_n$ be the sum of the first $n$ terms of a geometric sequence $\{a_n\}$. Given $3S_3 = a_4 - 3...

Sequences and SeriesGeometric SequenceCommon Ratio

Given an arithmetic sequence $\{a_n\}$, $S_n$ is the sum of the first $n$ terms. We know $S_{30} = 3...

Arithmetic SequencesSeriesSummation

Given a sequence $\{a_n\}$ with $a_1 = \frac{1}{2}$ and $a_{n+1} = 1 - \frac{1}{a_n}$, find $a_5$.

Sequences and SeriesRecursive FormulaIteration

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