Problems with the nth term

Two terms given

Often a problem gives two terms and asks for the AP. Write both with the formula, and subtract.

A worked example. The $3$rd term of an AP is $16$ and the $10$th term is $51$. Find the AP and its $20$th term.

1. Write both terms. $a + 2d = 16$ and $a + 9d = 51$.
2. Subtract. $7d = 35$, so $d = 5$.
3. Find $a$. $a = 16 - 2 \times 5 = 6$.
4. The AP. $6, 11, 16, 21, \dots$
5. The $20$th term. $a_{20} = 6 + 19 \times 5 = 101$.

Notice the quick idea in step 2: from the $3$rd term to the $10$th term is $10 - 3 = 7$ steps, so the jump $51 - 16 = 35$ is $7d$.

Three numbers in an AP

When a problem gives the sum of three numbers in an AP, write them as

The $d$'s cancel when you add, so the sum is just $3a$.

A worked example. Three numbers in an AP add up to $27$, and their product is $585$. Find them.

1. The sum. $(a - d) + a + (a + d) = 3a = 27$, so $a = 9$.
2. The product. $(9 - d) \times 9 \times (9 + d) = 585$, so $81 - d^2 = 65$.
3. Solve. $d^2 = 16$, so $d = 4$ or $d = -4$.
4. The numbers. $5, 9, 13$ (or the same numbers in reverse order).

Word problems

A situation gives an AP whenever something changes by the same amount at each step. Decide what $a$, $d$ and $n$ stand for, and keep the units.

Rows of seats. A hall has $18$ seats in the first row, and each row has $2$ seats more than the row in front. The $20$th row has $18 + 19 \times 2 = 56$ seats.

Savings. Arjun saves ₹120 in the first month and increases his savings by ₹30 every month. In which month does he save ₹570? Solve $120 + (n - 1) \times 30 = 570$: $(n - 1) \times 30 = 450$, so $n - 1 = 15$ and he does it in the $16$th month.

Counting numbers of a kind

How many two-digit numbers are multiples of $7$? They are $14, 21, 28, \dots, 98$, an AP with $a = 14$ and $d = 7$. The last term gives $14 + (n - 1) \times 7 = 98$, so $n - 1 = 12$ and there are $13$ of them.

Practise with new numbers

Try it in Python

This program lists the seats in each of the $20$ rows of the hall, so you can watch the AP grow.

Common mistakes

• Subtracting term numbers wrongly. From the $3$rd to the $10$th term is $7$ steps, not $10$.
• Writing three unknowns. Use $a - d$, $a$, $a + d$ so that the sum gives $a$ at once.
• Dropping a sign. $d^2 = 16$ gives $d = 4$ or $d = -4$; both give the same three numbers.
• Mixing up month $n$ and the number of increases. In month $16$ the extra amount has been added $15$ times.

Remember these

• Two given terms: subtract to get $d$, then find $a$.
• Three numbers in an AP: $a - d$, $a$, $a + d$. Their sum is $3a$.
• A word problem is an AP when the amount changes by the same step each time.
• To count numbers of a kind, find the first and the last, and solve for $n$.

Check yourself

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

1. The $2$nd term of an AP is $8$ and the $6$th term is $28$. Find $d$.

2. Three numbers in an AP have a sum of $36$. What is the middle number?

3. A taxi charges ₹50 for the first kilometre and ₹14 for each kilometre after it. What is the fare for $10\ \text{km}$?

4. How many two-digit numbers are multiples of $9$?

5. Why is $a - d, a, a + d$ a good way to write three numbers in an AP?