Find the sum of all multiples of 7 lying between 300 and 700.

$\dfrac{300}{7} = 42\dfrac{6}{7} \text{ and } \dfrac{700}{7} = 100$.

The numbers which are divisible by 7 between 300 and 700 are,

= 43 × 7, 44 × 7, 45 × 7, …………, 99 × 7.

= 301, 308, 315, ………., 693.

The above sequence is an A.P. with common difference = 7 and first term = 301 and last term = 693.

Let n be no. of terms,

∴ a_n = a + (n - 1)d

⇒ 693 = 301 + (n - 1)7

⇒ 693 = 301 + 7n - 7

⇒ 693 = 7n + 294

⇒ 693 - 294 = 7n

⇒ 7n = 399

⇒ n = 57.

$S = \dfrac{n}{2}(a + l) \\[1em] = \dfrac{57}{2} \times (301 + 693) \\[1em] = \dfrac{57}{2} \times 994 \\[1em] = 28329.$

Hence, sum = 28329.