# 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. - Mathematics

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

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#### APPEARS IN

RD Sharma Class 10 Maths
Chapter 5 Arithmetic Progression
Exercise 5.6 | Q 17 | Page 52