The Sum of an Arithmetic Progression: Quick Revision

A brief note on The Sum of an Arithmetic Progression. Read it before a test. Try each question at the end before you open its answer.

The sum formula

• Pair the first and last terms: every pair adds to $a + l$.
• $S_n = \dfrac{n}{2}(a + l)$, when the last term $l$ is known.
• $S_n = \dfrac{n}{2}[2a + (n - 1)d]$, when $a$ and $d$ are known.
• $1 + 2 + 3 + \cdots + n = \dfrac{n(n + 1)}{2}$.
• The sum of the first $n$ odd numbers is $n^2$.

Working back

• To find $n$: put the sum into the formula, solve the quadratic in $n$, and keep the natural-number roots.
• If both roots are natural numbers, both can be right: the terms in between add up to $0$.
• To find $a$ or $d$: put in what you know, and solve the linear equation.
• From a formula for the sum: $a_n = S_n - S_{n - 1}$, and $a_1 = S_1$.

In word problems

• A fixed rise or fall is the common difference $d$; a fall makes $d$ negative.
• "How much in month $n$" is a term. "How much in $n$ months" is a sum.

A worked example

How many terms of $12, 15, 18, \ldots$ add up to $255$?

1. The sum. $S_n = \dfrac{n}{2}[24 + 3(n - 1)] = \dfrac{n(3n + 21)}{2}$.
2. The equation. $3n^2 + 21n = 510$, so $n^2 + 7n - 170 = 0$.
3. Solve. $(n + 17)(n - 10) = 0$, so $n = 10$ or $n = -17$.
4. Choose and check. $n = 10$. The tenth term is $12 + 9 \times 3 = 39$, and $\dfrac{10}{2}(12 + 39) = 255$.

Always check a root by putting it back into the sum.

Practice with new numbers

A program to try

The program finds three sums by the formula and checks each one by adding the terms.

Mistakes to avoid

• Leaving out the half in $\dfrac{n}{2}$.
• Using $nd$ in place of $(n - 1)d$ for the last term.
• Keeping a negative or fractional $n$.
• Giving a term when a total is asked, or a total when a term is asked.

Check yourself

1. Find the sum of the first $20$ terms of $5, 8, 11, \ldots$

2. What is the sum of the even numbers from $2$ to $60$?

3. If $S_n = 5n^2 - 2n$, find the third term.

4. A tower of cups has $15$ cups at the bottom and one fewer in each row, up to $1$ cup at the top. How many cups are there?