P2.3 Sequences and series

How can a sequence be defined?

Every sequence question begins with the same questions. Is the sequence given by a formula for the \(n\)th term or by a rule linking each term to the one before? Does it go up, go down or repeat? Is it arithmetic (add \(d\)) or geometric (multiply by \(r\))? Once you know, a small set of formulae answers almost everything.

By a formula for the \(n\)th term

• The rule gives any term directly from its position: for \(u_n = n^2 + 1\), the terms are 2, 5, 10, 17, …, and \(u_{10} = 101\) without working out the terms in between.
• To test whether a number is a term, set the formula equal to it and check that \(n\) comes out as a positive integer. Is 145 a term of \(n^2 + 1\)? \(n^2 = 144\), so \(n = 12\): yes, it is the 12th term.

By a recurrence relation

• A recurrence relation gives each term from the one before: \(u_{n+1} = f(u_n)\). It needs a starting value, such as \(u_1 = 3\), or it does not define a single sequence.
• Generate terms one at a time by substituting: with \(u_{n+1} = 2u_n - 1\) and \(u_1 = 3\), you get 3, 5, 9, 17, …
• A recurrence with an unknown constant, such as \(u_{n+1} = ku_n - 3\), makes each term an expression in \(k\). A given term then gives an equation for \(k\).
• The same rule from a different starting value gives a different sequence, and possibly very different behaviour.

How do you tell if a sequence is increasing, decreasing or periodic?

• To prove a sequence is increasing, show that \(u_{n+1} - u_n > 0\) for every \(n\), not just for the first few terms.
• A sequence can be neither: \(u_n = n^2 - 4n + 7\) gives 4, 3, 4, 7, … Here \(u_{n+1} - u_n = 2n - 3\), which is negative for \(n = 1\) and positive for \(n \ge 2\).
• For a periodic sequence, generate terms until the first term reappears; the number of steps is the order (period). To add many terms, count the complete cycles and then the terms left over.

A recurrence can also settle down. With \(x_{n+1} = 0.5x_n + 3\) and \(x_1 = 1\) the terms are 1, 3.5, 4.75, 5.375, 5.6875, …: each one halves the gap to 6, so the sequence is increasing and gets closer and closer to 6. The staircase diagram shows why: move up to the line \(y = 0.5x + 3\) to get the next term, then across to \(y = x\) to use it as the next input.

What is an arithmetic sequence?

To reach the \(n\)th term from the first you take \(n - 1\) steps, not \(n\). The terms lie on a straight line when plotted against \(n\), with gradient \(d\): \(d > 0\) increasing, \(d < 0\) decreasing, \(d = 0\) constant.

• Three consecutive terms \(x, y, z\) are in arithmetic progression exactly when \(y - x = z - y\), i.e. \(2y = x + z\): the middle term is the mean of its neighbours.
• Two given terms give two linear equations in \(a\) and \(d\); subtract them to find \(d\) first.

How do you sum an arithmetic series, and why does the formula work?

A series is the sum of the terms of a sequence; \(S_n\) means the sum of the first \(n\) terms. The proof of the formula is short and you may be asked for it: write the sum forwards and backwards, then add.

The sum of the first \(n\) natural numbers. \(1 + 2 + 3 + \dots + n\) is arithmetic with \(a = 1\), \(d = 1\), \(l = n\), so \(\sum_{r=1}^{n} r = \frac12 n(n + 1)\). For example \(1 + 2 + \dots + 100 = 50 \times 101 = 5050\).

How does sigma notation work?

• \(\sum_{r=1}^{n} f(r)\) means: substitute \(r = 1, 2, 3, \dots, n\) into \(f(r)\) and add the results.
• The number of terms in \(\sum_{r=p}^{q}\) is \(q - p + 1\). From \(r = 11\) to \(r = 30\) there are 20 terms, not 19.
• If \(f(r)\) is linear in \(r\) the sum is arithmetic; if it is of the form \(A \times B^r\) it is geometric. Write out the first two or three terms to find \(a\) and \(d\) (or \(r\)).
• Useful rules: \(\sum_{r=1}^{n} c = cn\); \(\sum_{r=1}^{n} (pr + q) = p \times \frac12 n(n + 1) + qn\); \(\sum_{r=p}^{q} = \sum_{r=1}^{q} - \sum_{r=1}^{p-1}\).

What is a geometric sequence, and how do you sum a geometric series?

\(r > 1\) (with \(a > 0\)): increasing; \(0 < r < 1\): decreasing towards 0; \(r < 0\): the terms alternate in sign; \(r = -1\): periodic with order 2.

• Three consecutive terms \(x, y, z\) are in geometric progression exactly when \(\frac{y}{x} = \frac{z}{y}\), i.e. \(y^2 = xz\).
• Two given terms: divide one equation by the other to eliminate \(a\). For \(u_5\) and \(u_2\), you get \(r^3\).

The two forms are equal (multiply top and bottom by \(-1\)). Use the first when \(|r| < 1\) and the second when \(r > 1\), to keep the numbers positive.

When does a geometric series have a sum to infinity?

If \(|r| < 1\), then \(r^n \to 0\) as \(n \to \infty\), so \(S_n = \frac{a(1 - r^n)}{1 - r}\) gets closer and closer to \(\frac{a}{1 - r}\). The series is convergent. If \(|r| \ge 1\), the terms do not shrink and the sums do not settle: there is no sum to infinity (the series is divergent). The sum to infinity is a limit: no finite number of terms reaches it (for \(a \ne 0\) and \(r \ne 0\)), but \(S_n\) can be made as close to it as you like.

For Worked example 6: \(S_\infty = 150 \div 0.6 = 250\), and \(S_6\) is already within 1.024 of it.

How do you use logarithms to find \(n\)?

• Questions such as "the first term greater than 50 000" or "the least number of terms for the sum to exceed …" give an inequality with \(n\) in the power.
• Rearrange to the form \(r^n > k\) (or \(r^{n-1} > k\)), take logs of both sides and divide by \(\log r\). Any base works, as long as you use the same one on both sides.
• Direction of the inequality: if \(0 < r < 1\) then \(\log r < 0\), so dividing by it reverses the inequality. For example \(0.9^n < 0.01\) gives \(n\log 0.9 < \log 0.01\), so \(n > \frac{\log 0.01}{\log 0.9} = 43.7\), and \(n = 44\).
• \(n\) is a whole number. Give the smallest integer that satisfies the inequality, and check it and the integer below it if the critical value is close to a whole number.

Exam tips for P2.3

Common misconceptions

How this topic connects

P1 algebra and P2.1 proof

Questions on "the least \(n\) for which \(S_n > \dots\)" in arithmetic series become quadratic inequalities, solved with the methods from the P1 quadratics and inequalities pages. Both sum formulae are proofs by deduction.

P2 exponentials and logarithms, and P2.4 the binomial expansion

Finding \(n\) in a geometric sequence is solving \(a^x = b\); the binomial expansion is another way of writing out the terms of a finite series.

P3 numerical methods, P4 binomial series and IAL Physics

The staircase diagram is the same picture used for iteration \(x_{n+1} = g(x_n)\) in P3. In P4, \((1 - x)^{-1} = 1 + x + x^2 + \dots\) is the geometric series with sum to infinity \(\frac{1}{1 - x}\) for \(|x| < 1\). In physics, radioactive decay halves the number of nuclei every half-life, a geometric sequence with \(r = \frac12\).

Key formulae

Check your understanding

Specification checklist

• (P2 4.1) I can work out terms of a sequence from a formula for the \(n\)th term, and decide whether a given number is a term.
• (P2 4.1) I can generate a sequence from a recurrence relation \(u_{n+1} = f(u_n)\) and a first term, including one with an unknown constant.
• (P2 4.2) I can find the \(n\)th term of an arithmetic sequence, and find \(a\) and \(d\) from given information.
• (P2 4.2) I can prove the formula for the sum of the first \(n\) terms of an arithmetic series, and use it, including the sum of the first \(n\) natural numbers.
• (P2 4.2) I can read and evaluate sums written in sigma notation.
• (P2 4.3) I can decide whether a sequence is increasing, decreasing or periodic, and find the order of a periodic sequence and sums of its terms.
• (P2 4.4) I can find the \(n\)th term of a geometric sequence, and prove and use the formula for the sum of the first \(n\) terms of a geometric series.
• (P2 4.4) I know that a geometric series has a sum to infinity only when \(|r| < 1\), and I can find and use it.
• (P2 4.4) I can use logarithms to find the least value of \(n\) for which a term or a sum passes a given value.