Given a formula in n

Given a formula in $n$ (for $u_n$ or $S_n$ )

In 5 of 20 recent exams
Paper 1 or 2
3 to 7 marks

How it is set in the exam

Paper 1 (no calculator) or Paper 2 (calculator), 3–7 marks. Less common than the others, but students who haven't seen it usually substitute into the wrong formula. On Paper 2 (calculator) it is still algebra with $n$ as a letter; the GDC can only check a numerical answer.

Spot it in the exam

Exam phrases to spot (4)

“the sum of the first $n$ terms of a sequence is given by $S_n = \ldots$ ”
“the $n$ th term is $u_n = \ldots$ ”
“show that the sequence is arithmetic (geometric)”
“find an expression for $u_n$ ”

In disguise (4)

"find the 10th term" when only $S_n$ is given
$S_n = pn^2 + qn$ with $p$ and $q$ unknown, and two facts about the terms
an exponential in $n$ : $u_n = 5e^{0.3n}$ or $u_n = 5 \times 3^{2n-1}$ is geometric, but the ratio is not the number in front of $n$
a log of an exponential in $n$ : $u_n = \ln(2 \times 5^n)$ is arithmetic, because the log laws make it $\ln 2 + n\ln 5$ , linear in $n$

Looks like it, but isn't

a formula inside a $\sum$ with number limits. That is Sigma notation. A claim about every integer with no sequence in it ("show that … is a multiple of $8$ for all $n \in \mathbb{Z}$ ") is Prove it: from one side to the other (identities and integer claims).

Worked example

From $S_n$ to $u_n$

Paper 1 (no calculator)

Question

The sum of the first $n$ terms of a sequence is $S_n = 3n^2 + 2n$ .
(a) Find $u_1$ and $u_2$ . [3]
(b) Find an expression for $u_n$ . [3]

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Steps

Decide what is given: a term formula $u_n$ or a sum formula $S_n$ .
From $S_n$ :
$u_1 = S_1$ , and $u_n = S_n - S_{n-1}$ for $n \ge 2$ (the bigger sum minus the smaller leaves only the newest term). Substitute $n - 1$ with brackets. If $S_0 \neq 0$ (for a polynomial $S_n$ : a non-zero constant term), $u_1$ will not fit the formula.
Show arithmetic:
$u_{n+1} - u_n$ is a constant (no $n$ left). Show geometric: $\frac{u_{n+1}}{u_n}$ is a constant. Say "constant" (or "doesn't depend on $n$ ") in words: markschemes can give that statement a reasoning mark () of its own.
A log formula:
use the log laws first. $\ln(a \times b^n) = \ln a + n\ln b$ is linear in $n$ , so it is arithmetic with $d = \ln b$ ; then show it as in step 3. Find $u_1$ by substituting $n = 1$ : it is not the constant $\ln a$ .
Unknown constants in $S_n$
(such as $S_n = pn^2 + qn$ ): each fact gives one equation. At the end, state each constant (for example " $p = 2$ , $q = 5$ "): each one is its own mark, and rewriting only the $S_n$ formula loses it.
Check your $u_n$ against $u_1$ and $u_2$ worked out directly.

Common mistakes

$u_2 = S_2$ .
$S_2$ is $u_1 + u_2$ .
Dropping brackets
$S_{n-1}$ is $3(n-1)^2 + 2(n-1)$ , and the minus sign in front applies to all of it.
Using $u_n = S_n - S_{n-1}$ for $n = 1$ .
It needs $n \ge 2$ . If $S_0 \neq 0$ (for a polynomial, a non-zero constant term), $u_1$ differs from the formula's value. $S_n = 3^n - 1$ has $S_0 = 0$ , so there it does fit.

More mistakes (5)

Assuming arithmetic.
$S_n = 3^n - 1$ gives a geometric sequence. Check with $u_1$ , $u_2$ , $u_3$ .
Reading the ratio off the base
in $3^{2n-1}$ the ratio is $9$ .
Taking the number in front, or the constant, as $u_1$
$u_1$ comes from $n = 1$ . For $5 \times 3^{2n-1}$ it is $15$ , not $5$ ; for $\ln 2 + n\ln 5$ it is $\ln 10$ , not $\ln 2$ .
"Showing" with numbers
$u_2 - u_1 = u_3 - u_2$ is an example. The proof is $u_{n+1} - u_n$ with $n$ as a letter.
Not stating the constants
when $S_n$ has unknown constants, the answer is the constants (" $p = 2$ , $q = 5$ "). Writing only " $S_n = 2n^2 + 5n$ " loses the final mark.

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