in an AP, if the 5th and 12th terms are 30 and 65 respectively. what is the sum of first 20terms

Answer 1

An arithmetic progression (A.P) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.

If the initial term of an arithmetic progression is a1 and the common difference of successive members is d, then the nth term of the sequence is given by:

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

a 1 = first term

In this case:

a 5 = 30

a 1 + ( 5 - 1 ) d = 30

a 1 + 4 d = 30

a 1 = 30 - 4 d

a 12 = 65

a 1 + ( 12 - 1 ) d = 65

a 1 + 11 d = 65

30 - 4 d + 11 d = 65

30 + 7 d = 65

7 d = 65 - 30

7 d = 35 Divide both sides by 7

d = 35 / 7

d = 5

a 1 = 30 - 4 d

a 1 = 30 - 4 * 5

a 1 = 30 - 20

a 1 = 10

The sum S of the first n values of a finite sequence is given by the formula:

S n = ( n / 2 ) * [ 2 a 1 + ( n - 1 ) d ]

In tis case :

a 1 = 10

d = 5

n = 20

S 20 = ( 20 / 2 ) * [ 2 * 10 + ( 20 - 1 ) 5 ]

S 20 = 10 * ( 20 + 19 * 5 )

S 20 = 10 * ( 20 + 95 )

S 20 = 10 * 115

S 20 = 1150