Important Questions on Arithmetic Progression (Part 1)

This lesson works through three important Class 10 questions on arithmetic progressions (AP). Each one is solved step by step.

Question 1: Common difference from the nth term

We are given the nth term of an AP as

$a_n = 3n + 5$

and asked to find the common difference $d$.

When the nth term is written as a linear expression in $n$, the common difference is simply the coefficient of $n$. Here that coefficient is $3$, so

$d = 3$

Question 2: Finding terms from the sum formula

We are given the sum of the first $n$ terms as

$S_n = n^2$

and asked to find the common difference $d$.

First sum and first term. Put $n = 1$:

$S_1 = 1^2 = 1, \qquad a_1 = S_1 = 1$

Second sum. Put $n = 2$:

$S_2 = 2^2 = 4$

Second term. The second term is the second sum minus the first sum:

$a_2 = S_2 - S_1 = 4 - 1 = 3$

Common difference.

$d = a_2 - a_1 = 3 - 1 = 2$

Question 3: Counting the number of terms

We are given

$a_n = 500, \qquad a_1 = 50, \qquad d = 5$

and asked to find the number of terms $n$. Rearranging the nth term formula gives

$n = \frac{a_n - a_1}{d} + 1$

Substituting the values:

$n = \frac{500 - 50}{5} + 1 = \frac{450}{5} + 1 = 90 + 1 = 91$

So the AP has $91$ terms.

Key takeaways

• If $a_n = pn + q$, the common difference is the coefficient $p$.
• The first term equals the first sum, $a_1 = S_1$, and any term is $a_n = S_n - S_{n-1}$.
• The number of terms is $n = \dfrac{a_n - a_1}{d} + 1$.