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Question

Find the sum of all numbers from 1 to 140 which are divisible by 4.

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Solution

All the numbers from 1 to 140 that are divisible by 4 are 4, 8, 12, 16,…, 140.
Now, we observe that the numbers are in an A.P.
Here,
First term, a = 4
Common difference, d = t2 – t1 = 8 – 4 = 4
Let the number of these odd numbers be n.
We have: tn = 140
We know that the nth term of an A.P. is tn = a + (n – 1)d.
Thus, we have:
140 = 4 + (n – 1)4
140 – 4 = 4(n – 1)
136 = 4(n – 1)
(n – 1) = 34
n = 34 + 1 = 35
Thus, there are 35 numbers from 1 to 140 that are divisible by 4.
Now,
We know that the sum of n terms of an A.P is Sn=n22a +(n-1) d.
To find the sum of the numbers that are divisible by 4, we need to take a = 4, d = 4 and n = 35.
Thus, we have:

S35=3522×4+(35-1)4=3528+136 =352×144 =35×72 =2520

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