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In an A.P., If the 5th and 12th Terms Are 30 and 65 Respectively, What is the Sum of First 20 Terms? - CBSE Class 10 - Mathematics

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Question

In an A.P., if the 5th and 12th terms are 30 and 65 respectively, what is the sum of first 20 terms?

Solution

In the given problem, let us take the first term as a and the common difference d

Here, we are given that,

`a_5 = 30`    ....(1)

`a_12 = 65` .....(2)

Also, we know

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

For the 5th term (n = 5)

`a_5 = a + (5 - 1)d`

30 = a + 4d       (Using 1)

a = 30 - 4d ....(3)

Similarly for the 12 th term (n = 12)

`a_12 = a + (12 - 1)d`

65 = a + 11d     (Using 2)

a = 65 - 11d....(4)

Substracting (3) from (4) we get

a - a = (65 - 11d)-(30 - 4d)

0 = 65 - 11d - 30 + 4d

0 = 35 - 7d

7d = 35

d = 5

Now, to find a, we substitute the value of d in (4),

a = 30 - 4(5)

a = 30 - 20

a = 10

So for the given A.P d = 5 and a = 10

So to find the sum of first 20 terms of this A.P. we use the following formula for the sum of n terms of an A.P

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

Where a = first term for the given A.P

d= common difference of the given A.P

n = number of terms

So using the formula for n = 20 we get

`S_20 = 20/2 [2(10) + (20 - 1)(5)]`

= (10)[20 + (19)(5)]

= (10)[20 + 95]

= (10)[115]

= 1150

Therefore, the sum of first 20 terms for the given A.P. is `S_20 = 1150`

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Solution In an A.P., If the 5th and 12th Terms Are 30 and 65 Respectively, What is the Sum of First 20 Terms? Concept: Sum of First n Terms of an AP.
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