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If 12th Term of an A.P. is −13 and the Sum of the First Four Terms is 24, What is the Sum of First 10 Terms. - CBSE Class 10 - Mathematics

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Question

If the 12th term of an A.P. is −13 and the sum of the first four terms is 24, what is the sum of first 10 terms?

Solution

In the given problem, we need to find the sum of first 10 terms of an A.P. Let us take the first term a and the common difference as d

Here, we are given that,

`a_12 = -13`

`S_4 = 24`

Also, we know

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

For the 12th term (n = 12)

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

`-13 = a + 11d`

a= -13 - 11d ......(1)

 So, as we know the formula for the sum of n terms of an A.P. is given by,

`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.

= number of terms

So, using the formula for n = 4, we get,

`S_4  = 4/2 [2(a) + (4 - 1)(d)]`

24 = (2)[2a + (3)(d)]

24 = 4a +6d

4a  = 24 - 6d

` a= 6 - 6/4 d` ....(2)

Subtracting (1) from (2), we get,

`a - a =  (6 - 6/4 d) - (-13 - 11)d`

`0 = 6 - 6/4 d + 13 + 11d`

`0 = 19 + 11d - 6/4 d`

`0 =19+ (44d - 6d)/4`

On further simplifying for d, we get,

`0 = 19 + (38d)/4`

`-19=19/2 d`

`d= (-19(2))/2`

d= -2

Now, to find a, we substitute the value of d in (1),

a = -13 -11(-2)

a = -13 + 22

a = 9

Now using the formula for the sum of n terms of an A.P. for n = 10 we get

`S_10 = 10/2 [2(9) + (10 - 1)(-2)]`

= (5)[18 + (9)(-2)]

= (5)(18 - 18)

=(5)(0)

= 0

Therefore the sum of first 10 terms for the given A.P is `S_10 = 0`

  Is there an error in this question or solution?

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Solution If 12th Term of an A.P. is −13 and the Sum of the First Four Terms is 24, What is the Sum of First 10 Terms. Concept: Sum of First n Terms of an AP.
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