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# Question 55. Find the sum of all integers(v) from 1 to 500 which are multiples of 2 or 5.

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This is an arithmetic progression based question from Chapter name- Arithmetic Progression
Chapter number- 9
Exercise – 9.6
In this question we have to Find the sum of all integers from 1 to 500 which are multiples of 2 or 5.

CBSE DHANPAT RAI PUBLICATIONS
Understanding CBSE Mathematics
Class :- 10th
Question no 55(v)

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1. Integers from 1 to 500 which are multiples of 2 are 2, 4, 6, 8, . . . .500.

First term(a) = 2, common difference(d) = 4 – 2 = 2 and nth term(an) = 500.

By using the formula of nth term of an A.P.

an = a + (n – 1)d

So,

500 = 2 + (n – 1)2

=> 2(n–1) = 498

=> n–1 = 249

=> n = 250

Let S1 be the sum of this A.P. Hence, S1 = 250[2 + 500] / 2 = 125 = 62750.

Integers from 1 to 500 which are multiples of 5 are 5, 10, 15, 20, . . . .500.

First term(a) = 5, common difference(d) = 10 – 5 = 5 and nth term(an) = 500.

By using the formula of nth term of an A.P.

an = a + (n – 1)d

So,

500 = 5 + (n – 1)5

=> (n – 1)5 = 495

=> n – 1 = 99

=> n = 100

Let S2 be the sum of this A.P. Hence, S2 = 100[5 + 500] / 2 = 50 = 25250.

Integers from 1 to 500 which are multiples of 2 as well as 5 are 10, 20, 30 . . . .500.

We know, 500 = 10 + (n – 1)10

=> 10(n – 1) = 490

=> n – 1 = 49

=> n = 50

Let S3 be the sum of this A.P. Hence, S3 = 50[10 + 500] / 2 = 25 = 12750.

Hence, required sum = S1 + S2 – S3

= 62750 + 25250 – 12750

= 75250

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