Working back from the sum

How many terms?

How many terms of $9, 17, 25, \ldots$ must be added to get $636$?

Here $a = 9$, $d = 8$, and $n$ is unknown. Put them in the formula:

1. Simplify. $\dfrac{n}{2}[8n + 10] = 636$, so $n(4n + 5) = 636$.
2. Make a quadratic equation. $4n^2 + 5n - 636 = 0$.
3. Solve. The discriminant is $25 + 4 \times 4 \times 636 = 10201 = 101^2$, so $n = \dfrac{-5 \pm 101}{8}$.
4. Choose. $n = 12$ or $n = -\tfrac{53}{4}$. A number of terms must be a natural number, so $n = 12$.

Check. $S_{12} = 6[18 + 88] = 6 \times 106 = 636$.

When both answers make sense

Sometimes both roots are natural numbers, and both are right. Take $15, 12, 9, 6, \ldots$ and ask how many terms give $45$.

So $n = 5$ or $n = 6$. Both work, because the sixth term is $15 - 5 \times 3 = 0$. Adding a zero leaves the sum at $45$.

Finding the first term or the common difference

The sum of the first $12$ terms of an arithmetic progression is $246$, and its common difference is $3$. Find the first term.

The progression is $4, 7, 10, \ldots$

Terms from a formula for the sum

Suppose $S_n = 2n^2 + 3n$ for every $n$. The sum of the first $n$ terms is the sum of the first $n - 1$ terms, plus the $n$th term. So

• The first term is $S_1 = 2 + 3 = 5$.
• The second term is $S_2 - S_1 = 14 - 5 = 9$.
• In general, $a_n = (2n^2 + 3n) - (2(n - 1)^2 + 3(n - 1)) = 4n + 1$.

So the progression is $5, 9, 13, \ldots$, with common difference $4$.

Practise with new numbers

Try it in Python

The program adds the terms of $9, 17, 25, \ldots$ one at a time, until the sum reaches $636$.

Common mistakes

• Keeping a root that is not a natural number. The number of terms is never negative or a fraction.
• Throwing away a root too quickly. If both roots are natural numbers, check both in the progression.
• Taking $S_n$ for $a_n$. $S_n$ is a total. The $n$th term is $S_n - S_{n - 1}$.
• Forgetting $S_1$. The first term is $S_1$ itself, since $S_0 = 0$.

Remember these

• To find $n$, put $S_n$ into the formula and solve the quadratic equation in $n$.
• Keep only the roots that are natural numbers, and check them.
• $a_n = S_n - S_{n - 1}$, and $a_1 = S_1$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. How many terms of $1, 3, 5, \ldots$ add up to $169$?

2. The sum of the first $8$ terms of an arithmetic progression is $100$, and the first term is $2$. Find the common difference.

3. If $S_n = n^2 + 6n$, what is the first term?

4. With the same $S_n = n^2 + 6n$, find the common difference.

5. Ria solves a quadratic for $n$ and gets $n = 7$ or $n = -9$. Which does she keep?