Question

Question 1

The sum of the first ten terms of an Arithmetic
Progression (A.P.) is 130. If the fifth term is 3
times the first term, find the:
a. common difference;
b. first term;
c. number of terms of the A.P. if the last term is
28.​

Answer 1

a. 2

b. 4

c. 13

Explanation:

The general term of an arithmetic progression is ...

an = a1+d(n-1)

where a1 is the first term, and d is the common difference.

The sum of n terms is ...

Sn = n(2a1 +d(n -1))/2

__

The given relations tell us ...

S10 = 10(2a1 +9d)/2 = 10a1 +45d = 130

and

a5 = 3a1

a1 +4d = 3a1

4d = 2a1

2d = a1

Using this in the equation for S10 above, we have ...

10(2d) +45d = 130

d = 130/65 = 2

___

(a) The common difference is 2.

d = 2

__

(b) The first term is a1 = 2d = 2(2)

a1 = 4.

__

(c) an = 28 = 4 +2(n -1)

24 = 2(n -1)

12 = n -1

13 = n

The 13th term is 28.