Question

Find the number of terms of the A.P. If 1 is added to each term of this A.P., then find the sum of all terms of the A.P. thus obtained.

Answer 1

The number of terms of A.P. is 12. The sum of all terms of the A.P. thus obtained is 66.

The complete question is,

Find the number of terms of the A.P -12, -9, -6,... 21. If 1 is added to each term of this A.P., then find the sum of all terms of the A.P. thus obtained.

Given,

The A.P series

-12, -9, -6,... 21

a = -12

d = -9 - (-12) = -9 + 12 = 3

an = 21

we use a formula an = a + (n-1) d

21 = -12 + (n-1) 3

21 + 12 = 3n - 3

33 + 3 = 3n

36 = 3n

∴ n = 12

Sum of terms of the A.P.

Sn = n/2 [ a + an ]

S(12) = 12/2 [ -12 + 21 ]

= 6 [ 9 ]

∴ S(12) = 54

If 1 is added to each term of this A.P, the sum of all the terms of the new A.P. will increase be n = 12.

The new sum is,

Sn = 54 + 12 = 66.

Verification:

The new series becomes,

-12+1, -9+1, -6+1,... 21+1.

-11, -8, -5,.........22

a = -11, d = -8 + 11 = 3 an = 22

S(n) = n/2 [ a + an ]

66 = n/2 [ -11 + 22 ]

132 = n [ 11 ]

n = 12