Sigma notation

Find the value, or find the unknown

In 2 of 20 recent exams
Paper 1 or 2
3 to 6 marks

How it is set in the exam

Paper 1 (no calculator) or Paper 2 (calculator), 3–6 marks. Rare as a question of its own. Sigma notation turns up more often as notation inside other types: $S_n$ written as a sigma, or a sum to infinity written as $\sum_{r=1}^{\infty}$ . (On Paper 2 (calculator), a sum with an unknown number of terms is solved as in Find n.)

Spot it in the exam

Exam phrases to spot (3)

“find the value of $\sum_{r=1}^{20} (3r - 1)$ ”
“given that $\sum \ldots = 3066$ , find $n$ ”
“write the series in sigma notation”

In disguise (8)

a bottom limit that isn't $1$ (the number of terms changes)
a coefficient in front of a power, $3 \times 2^k$ , which is not the first term
the unknown is the top limit, a constant inside the bracket, or a factor in front
"write the series $4 + 11 + 18 + \cdots + 144$ in sigma notation"
a sum that doesn't start at the first term: "the sum of the 11th to the 30th terms" is $S_{30} - S_{10}$
sigma used as notation for $S_n$ : " $S_n = \sum_{r=1}^{n} 3(0.6)^r$ ". Decode $u_1 = S_1$ and the ratio, then the question is really about $S_n$ or $S_\infty$ (it usually carries on as Sum to infinity or Find n)
a term that is neither arithmetic nor geometric, such as $2^k - 1$ (it turns up when the terms being added are themselves partial sums, here of $1 + 2 + 4 + \cdots$ ): split it into standard series, $\sum (2^k - 1) = \sum 2^k - \sum 1$ , where $\sum_{k=1}^{n} 1 = n$
an infinite top limit: $\sum_{r=1}^{\infty}$ is a sum to infinity

Looks like it, but isn't

a $\sum$ of logs whose terms cancel, such as $\sum_{k=1}^{n} \ln\frac{k+1}{k}$ : Log laws: combine, split, or use given log values (telescoping).

Worked example

Geometric, the top limit is unknown

Paper 1 (no calculator)

Question

Given that $\displaystyle\sum_{k=1}^{n} 3 \times 2^k = 3066$ , find the value of $n$ . [5]

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Steps

Substitute the bottom limit, then the next value, to get the first two terms.
Decide the type: counter to the power 1 → arithmetic; counter in an exponent → geometric, with ratio $\frac{u_2}{u_1}$ . Neither → split the term into parts that are (step 6).
Number of terms = top − bottom + 1.
Apply the $S_n$ formula with that first term and that number of terms. On the GDC can evaluate the sum directly; write down the sum you evaluated.
If there is an unknown, set the sum equal to the given value and solve. An unknown top limit is solved as in Find n.
A non-standard term: split the sum into standard sums (arithmetic, geometric, and a constant counted once per term), sum each one, then add.

Common mistakes

Counting terms as top − bottom.
$\sum_{r=5}^{24}$ has 20 terms, not 19.
Taking the coefficient as $u_1$
in $\sum_{k=1}^{n} 3 \times 2^k$ the first term is $6$ .
Starting at $0$
$\sum_{n=0}^{9}$ has 10 terms and its first term uses $n = 0$ (so $2^0 = 1$ ).

More mistakes (2)

Taking the common difference from the constant
in $4r - 3$ , $d = 4$ , not $-3$ .
A constant inside the bracket counted once
$\sum_{r=1}^{10} (5 + 2r)$ contains $5$ ten times, so $50$ , not $5$ .

Easy to confuse with

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