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Find the sum of those integers from 1 to 500 which are multiples of 2 or 5. [Hint : These numbers will be multiples of 2 + multiples of 5 - multiples of 2 as well as of 5 ]

Solution:

Given, the series is a multiple of 2 or 5.

We have to find the sum of the integers from 1 to 500 which are multiples of 2 or 5.

a) The series which is a multiple of 2 is 2, 4, 5,........,500

The nth term of the series in AP is given by

aₙ = a + (n - 1)d

Here, a = 2, d = 4 - 2 = 2

500 = 2 + (n - 1)(2)

500 - 2 = (n - 1)(2)

498 = (n - 1)2

n - 1 = 498/2

n - 1 = 249

n = 249 + 1

n = 250

If l is the last term of an AP, then the sum of the terms is given by

S = n/2[a+l]

So, S₂₅₀ = 250/2[2 + 500]

= 125[502]

S₂₅₀ = 62,750

b) The series which is a multiple of 5 is 5, 10, 15,........,500

The nth term of the series in AP is given by

aₙ = a + (n - 1)d

Here, a = 5, d = 10 - 5 = 5

500 = 5 + (n - 1)(5)

500 - 5 = (n - 1)(5)

495 = (n - 1)5

n - 1 = 495/5

n - 1 = 99

n = 99 + 1

n = 100

If l is the last term of an AP, then the sum of the terms is given by

S = n/2[a+l]

So, S₁₀₀ = 100/2[5 + 500]

= 50[505]

S₁₀₀ = 25,250

C) The series which is a multiple of both 2 and 5.

Taking the LCM of 2 and 5 = 10

The series is a multiple of 10.

The series is 10, 20, 30,........500

The nth term of the series in AP is given by

aₙ = a + (n - 1)d

Here, a = 10, d = 20 - 10 = 10

500 = 10 + (n - 1)(10)

500 - 10 = (n - 1)(10)

490 = (n - 1)10

n - 1 = 490/10

n - 1 = 49

n = 49 + 1

n = 50

If l is the last term of an AP, then the sum of the terms is given by

S = n/2[a+l]

So, S₅₀ = 50/2[10 + 500]

= 25[510]

S₅₀ = 12,750

Now, sum of the integers which is a multiple of 2 or 5 is sum of multiples of 2 + sum of multiples of 5 - sum of multiples of 2 as well as of 5

= S₂₅₀ + S₁₀₀ - S₅₀

= 62750 + 25250 - 12750

= 88000 - 12750

= 75250

Therefore, the sum of the integers from 1 to 500 which is a multiple of 2 or 5 is 75250.

✦ Try This: Find the sum of those integers from 500 to 900 which are multiples of 2 or 5

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 5

NCERT Exemplar Class 10 Maths Exercise 5.4 Problem 2 (iii)

Find the sum of those integers from 1 to 500 which are multiples of 2 or 5

Summary:

The sum of those integers from 1 to 500 which are multiples of 2 or 5 is 75250.

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