Series

Learn, then practise

The original worked lessons remain the main explanations. Use these focused reminders to check the conditions before you start.

Arithmetic progressions: terms, sums and integer thresholds

Recover the first term and common difference from nonconsecutive terms.
Distinguish an individual term from a finite or partial sum.
Retain every positive integer solution and check threshold neighbours.

For an AP, \(u_n=a+(n-1)d\) and \(\displaystyle S_n=\frac n2[2a+(n-1)d]\). The sum from term \(p\) to term \(q\) has \(q-p+1\) terms. Indices are positive integers: test neighbouring values rather than simply rounding.

Read the full notes and 3 worked examples · Next: short practice

Geometric progressions: finite sums, convergence and remainders

Recover all possible ratios, including negative ones.
Distinguish finite sums from a convergent infinite sum.
Handle exceptional ratios and use the absolute omitted tail for error bounds.

Use \(u_n=ar^{n-1}\) and \(\displaystyle S_n=\frac{a(1-r^n)}{1-r}\) when applicable; for \(r=1\), \(S_n=na\). If \(a\ne0\), an infinite sum exists exactly when \(|r|<1\), and \(\displaystyle S_\infty=\frac a{1-r}\). The signed remainder is \(R_n=S_\infty-S_n\); an error tolerance uses \(|R_n|\). For \(r=0\), write \(a,0,0,\ldots\) directly.

Read the full notes and 3 worked examples · Next: short practice

Trigonometric terms in an arithmetic progression

Translate consecutive AP terms into twice-middle equals outer-sum.
Use P1 identities while preserving zero cases and every angle in the stated interval.
Find the difference and extend the selected progression.

For consecutive AP terms \(u,v,w\), use \(2v=u+w\). Zero terms and \(d=0\) are allowed. Solve on the stated interval and recover \(d=v-u\) for each angle; do not divide by a trigonometric factor without checking its zeros.

Read the full notes and 3 worked examples · Next: short practice

Trigonometric terms in a geometric progression

Use the GP product condition with recurrence checks for zero terms.
Retain both ratio signs when the question permits them.
Apply positivity, angle intervals and convergence separately.

For nonzero consecutive terms \(u,v,w\), \(v^2=uw\) gives the common-ratio test. When zeros are allowed, check \(v=ru\) and \(w=rv\) for a finite real \(r\). This package allows \(r=0\) unless a question says otherwise. The product test alone can accept a false candidate such as \((0,0,8)\).

Read the full notes and 3 worked examples · Next: short practice

Series models: accumulation, decay and repeated motion

Choose an additive or multiplicative model and state the first index.
Distinguish an amount remaining, a total used and a signed displacement.
Account for initial events, model changes and strict integer thresholds.

Label the initial event before applying a formula. A starting amount at stage 0 usually produces a power \(r^n\) at stage \(n\). In a rebound model, count the initial fall once and each completed rebound height twice. A sum of geometric areas uses the square of the linear scale factor.

Read the full notes and 3 worked examples · Next: short practice