Question

Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.

Answer 1

sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667

• all odd integers between 1 and

1000 which are divisible by 3 are

3,9,15,21.........993,999

• clearly, it's an AP with a=3 and d=6 and L=999

• An=a+(n-1)d

•999=3+(n-1)(6)

•996=(n-1)(6)

•166=n-1

•n=167

•Now, Sn=(n/2)[a+L]

• Sn = (167/2)(3+999)

• Sn = 167×1002÷2

• Sn = 167×501

• Sn = 83667

Answer 2

The sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667, shown.

Step-by-step explanation:

The all odd integers between 1 and 1000 which are divisible by 3:

3, 9, 15, 21, ........., 999

The given series are in A.P.

Here, first term (a) = 3, common difference (d) = 9 - 3 = 6 and

last term (l) = 999

Let n be the number of terms.

We know that,

The nth term of an A.P.

$a_{n}$ = a + (n - 1)d

⇒ 3 + (n - 1)6 = 999

⇒ (n - 1)6 = 999 - 3 = 996

⇒ n - 1 = 166

⇒ n = 166 + 1 = 167

∴ Sum = $\dfrac{n}{2} (a+l)$

= $\dfrac{167}{2} (3+999)$

= $\dfrac{167}{2} (1002)$

= 167 × 501

= 83667

Thus, the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667, shown.