FINDING THE SUM OF GIVEN NUMBER OF TERMS FROM THE GIVEN INFORMATION

Question 1 :

Given a = 7, a₁₃ = 35 find d and S₁₃  

Solution :

13^th term of the arithmetic progression,

a₁₃  =  35

a + 12d  =  35 ---(1)

By applying the value of a in (1), we get d

7 + 12d  =  35

12d  =  35 - 7

12d  =  28

d  =  28/12  =  14/6  =  7/3

S_n  =  (n/2) [2a + (n - 1)d]

S₁₃  =  (13/2) [2(7) + (13 - 1)(7/3)]

S₁₃  =  (13/2) [14 + 12(7/3)]

S₁₃  =  (13/2) [14 + 28]

S₁₃  =  (13/2) (42)

S₁₃  =  273

Hence, the sum of 13 terms of the arithmetic progression is 273.

Question 2 :

Given a₁₂ = 37, d = 3 find a and S₁₂ 

Solution :

12^th term of AP :

a₁₂ = 37

a + 11d  =  37 -----(1)

By applying the value of d in (1), we get

a + 11(3)  =  37

a + 33  =  37

a  =  37 - 33  =  4

Now, we find the sum of 12 terms

s_n  =  (n/2)[2a + (n - 1|)d]

s₁₂  =  (12/2)[2(4) + (12 - 1)(3)]

s₁₂  =  6[8 + 11(3)]

s₁₂  =  246

Hence, the sum of first 12 terms is 246.

Question 3 :

Given a₃ = 15 , S₁₀ = 125 find d and a₁₀  

Solution :

Here, we have 3^rd term and sum of first 10 terms. By using these information, we should find the common difference and 10^th term.

a₃ = 15

a + 2d  =  15  ----(1)

S₁₀ = 125

S_n  =  (n/2)[2a + (n - 1)d]

Here, n = 10

(10/2)[2a + (10 - 1)d]  =  125

5[2a + 9d]  =  125

Dividing each side by 5.

2a + 9d  =  25  ----(2)

By solving (1) and (2), we will get the values of a and d.

(1) ⋅ 2 - (2)

2a + 4d  =  10

2a + 9 d  =  25

(-)     (-)     (-) 

--------------------

-5d  =  -15

d  =  3

By applying the value of d in (1), we get

a + 2(3)  =  15

a + 6  =  15

a  =  15 - 6  =  9

10^th term :

a₁₀  =  a + 9d

a₁₀  =  9 + 9(3)

=  9 + 27

a₁₀  =  36

Hence, the 10^th term is 36.

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