Q) Find the sum of all integers between 50 and 500, which are divisible by 7.

Ans: 

Step 1: Let’s find out 1st term and last term.

Since we need to find out numbers divisible by 7 between 50 and 500. These numbers will be in an AP where common difference will be 7.

1st term will be the number which is higher than 50 and divisible by 7 i.e. 56

Last term of this AP is the number which is smaller than 500 and divisible by 7 i.e. 497

Step 2: Calculate number of terms in the AP:

We know that the nth term of an AP is given by, T_n  = a + (n – 1) d

Where, a = first term of the AP

d = common difference of the AP

Hence, n^th term, 497 = 56 + ( n – 1) 7

∴ 497 – 56 = (n – 1) 7

∴ 441 = (n – 1) 7

∴ (n -1) =

∴ n – 1 = 63

∴ n = 64

Therefore there will be 64 terms of this AP (i.e. there will be 64 numbers between 50 and 500 which are divisible by 7)

Alternatively, Total numbers divisible by 7 between 0 to 500 = 71

Similarly, total numbers divisible by 7 between 0 to 50 = 7

So, total numbers between 50 and 500 which are divisible by 7 are: 71 – 7 = 64

Step 3: Calculating sum of these 64 terms:

We know that the sum of n terms is given by,

∴

∴ = 32 (112 + 63 x 7)

∴ = 32 (553) = 17,696

Therefore, sum of all integers is 17, 696.

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