Question

Find sum of all 3 digit numbers which are not divisible by 7.

Answer 1

Sangeetac, hello

To solve the question you are asking, just use the formula for the sum of an arithmetic progression.  Use it to sum all the three digit numbers from 100 to 999.  Then subtract from that the sum the sum of all three digit numbers divisible by 7 (105 through 994).

The sum of all 3 digit numbers which are not divisible by 7 is: 424,214.

Answer 2

Sum of all three digit number which are not divisible by 7 = sum of all three digit numbers - sum of all three digit numbers divisible by 7

Sum of all three digit numbers,

100 + 101 + 102..... + 999

a_n = a + (n - 1)d

999 = 100 + (n - 1) x 1

999 - 100 = n - 1

899 + 1 = n

900 = n

Sum of all three digit numbers (S₁)

so we know that,

S₁ = n/2 (a + l)

S₁ = 900/2 (100 + 999)

S₁ = 450 x 1099

S₁ = 494550

Now, sum of all three digit numbers divisible by 7 (S₂)

105 + 112...... + 994

a_n = a + (n - 1)d

994 = 105 + (n - 1) x 7

994 - 105 = (n - 1) x 7

889/7 = n - 1

127 + 1 = n

128 = n

Now,

S₂ = n/2 (a + l)

S₂ = 128/2 (105 + 994)

S₂ = 64(1099)

S₂ = 70336

Now, sum of all three digit numbers not divisible by 7 = sum of all three digit numbers - sum of all three digit numbers divisible by 7

= 494550 - 70336

= 424214

Therefore, sum of all three digit numbers not divisible by 7 is 424214.