Sequences and Series

Sequences versus series

A sequence lists terms one at a time. A series adds them together. Finding the sum of the first $n$ terms of a sequence, instead of just one individual term, requires a dedicated sum formula. Using the term formula in its place is one of the most common errors on this topic. Before you touch a formula, ask yourself two questions: am I being asked for one term or for a total? and is the pattern a constant difference or a constant ratio?

Arithmetic series

An arithmetic series sums a sequence with a constant common difference $d$. Its sum is

$S_n = \dfrac{n}{2}\left(2a_1 + (n-1)d\right)$

or equivalently $S_n = \dfrac{n}{2}(a_1 + a_n)$ once the last term is known. The second form says: the sum is the number of terms times the average of the first and last terms. If you are given the last term but not $n$, find $n$ first from $a_n = a_1 + (n-1)d$, and count carefully. An off-by-one error in $n$ is the most frequent way to miss these questions.

Worked Example. Find the sum of the first 9 terms of the arithmetic sequence with $a_1 = 6$ and $d = 3$.

Apply the formula directly:

$S_9 = \dfrac{9}{2}\left(2(6) + 8(3)\right) = \dfrac{9}{2}(12 + 24) = \dfrac{9}{2}(36) = 162$

Notice that $(n-1) = 8$ multiplies $d$, while $n = 9$ appears only in the factor $\frac{n}{2}$. Mixing those up is the most common slip.

Geometric series

A geometric series sums a sequence with a constant common ratio $r$. Its sum is

$S_n = \dfrac{a_1(1 - r^n)}{1 - r}, \quad r \ne 1$

Worked Example. Find the sum of the geometric series $5 + 10 + 20 + 40 + 80$.

This is a 5-term geometric series with $a_1 = 5$ and $r = 2$:

$S_5 = \dfrac{5(1 - 2^5)}{1 - 2} = \dfrac{5(-31)}{-1} = 155$

When the ratio is negative, keep the sign inside the power: with $r = -2$, the quantity $r^5$ is $-32$, not $32$, and $1 - r$ becomes $3$, not $-1$.

Sigma notation

Sigma notation, $\Sigma$, is shorthand for a sum written out term by term. $\displaystyle\sum_{k=1}^{n} f(k)$ means: substitute $k = 1$, then $k = 2$, and so on through $k = n$ into the expression $f(k)$, and add the results. Recognizing sigma notation as nothing more than a compact way to write a sum removes most of its difficulty.

Worked Example. Evaluate $\displaystyle\sum_{k=1}^{4} (2k + 5)$.

Substitute each value of $k$ from 1 to 4: $7,\ 9,\ 11,\ 13$. Then $7 + 9 + 11 + 13 = 40$.

Many sums written this way are arithmetic or geometric series once expanded, but not all are. A sum like $\sum k^2$ follows neither pattern, so when no familiar series structure appears, expand the indicated terms directly instead of forcing a formula that does not apply. Pay attention to where the index starts. $\sum_{k=3}^{7} k^2$ is $9 + 16 + 25 + 36 + 49$, not the sum of the first five squares.

A full worked example

Worked Example. A sequence begins $2, 6, 18, 54$. Find the sum of the first 6 terms.

Recognize. Each term is 3 times the previous term, a constant ratio instead of a constant difference. This is a geometric series, not an arithmetic one, so the arithmetic sum formula does not apply.

Set up. $a_1 = 2$, $r = 3$, $n = 6$, so $S_6 = \dfrac{a_1(1 - r^6)}{1 - r}$.

Solve. $S_6 = \dfrac{2(1 - 729)}{1 - 3} = \dfrac{2(-728)}{-2} = 728$.

Check. Listing all six terms directly: $2, 6, 18, 54, 162, 486$. Their sum is $728$, matching the formula.

Best route. The formula by hand. A calculator helps evaluate $3^6$ quickly but does not decide which series formula applies.

Common error. Applying the arithmetic series formula using the first difference ($6 - 2 = 4$) as if it were constant, instead of recognizing the constant ratio.

Where this appears as a model

A savings plan that increases by a fixed amount each period is an arithmetic series. Saving $50 in the first month and increasing by $10 each month after that totals $1,260 after 12 months, found with the arithmetic series formula instead of adding twelve terms individually. Rows of seats that grow by a fixed number, or a stack of logs losing one log per row, work the same way. A quantity that is multiplied by a fixed percentage each time, like a bouncing ball reaching 60% of its previous height, is geometric.

Strategy checklist

• Classify first: constant difference means arithmetic, constant ratio means geometric, neither means expand by hand.
• Count the terms: if you know the last term, solve $a_n = a_1 + (n-1)d$ for $n$ before summing.
• Use the right formula: $(n-1)$ goes with $d$ or in the exponent count, and $n$ goes in $\frac{n}{2}$ or $r^n$.
• Sigma: note the starting and ending index, then substitute and add.
• Check: for small $n$, list the terms and add them to confirm.