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Find the Sum of All Integers Between 50 and 500, Which Are Divisible by 7. - CBSE Class 10 - Mathematics

ConceptSum of First n Terms of an AP

Question

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

Solution

In this problem, we need to find the sum of all the multiples of 7 lying between 50 and 500.

So, we know that the first multiple of 7 after 50 is 56 and the last multiple of 7 before 500 is 497.

Also, all these terms will form an A.P. with the common difference of 7.

So here,

First term (a) = 56

Last term (l) = 497

Common difference (d) = 7

So, here the first step is to find the total number of terms. Let us take the number of terms as n.

Now, as we know,

a-n = a+ (n -1)d

So, for the last term,

497 = 56 + (n -1)7

497 = 56+ 7n - 7

497 = 49 +7n

497 - 49 = 7n

Further simplifying

448 = 7n

n = 448/7

n = 64

Now, using the formula for the sum of n terms,

S_n = n/2 [2a + (n -1)d]

For n = 64 we get

S_n = 64/2 [2(56) + (64 - 1) 7]

= 32 [112 + (63)7]

= 32(1123 + 441)

= 32(112 + 441)

= 32(553)

= 17696

Therefore the sum of all the multiples of 7 lying betwenn 50  and  500 is S_n = 17696

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Solution Find the Sum of All Integers Between 50 and 500, Which Are Divisible by 7. Concept: Sum of First n Terms of an AP.
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