# The sum of the 1st 4 terms of an AP is 216 and the sum of the 1st 18 terms is 988. What is the first term.

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### 2 Answers

The sum of the first n terms of an arithmetic progression with first term a and common difference d is `S_n = (n/2)*(2a + (n - 1)*d)`

Here, S4 = 2*(2a + 3d) = 216 and S18 = 9*(2a + 17d) = 988

=> 2a + 3d = 108 and 2a + 17d = 988/9

=> 34a + 51d = 1836 and 6a + 51d = 988/3

Subtract the two equations

=> 34a - 6a = 1836 - 988/3

=> 28a = 4520/3

=> a = 1130/21

**The first term of the series is 1130/21**

SO FIRst of all

t4 = a +(n-1)d

216= a +(4-1)d

216 = a + 3d

so now solve for "a"

you get

a = 216 - 3d

now second case

t18 = a +(18-1)d

988= a+ 17d

now subsitute the value of a as you found above ,

988 = 216-3d +17d .... 988-216 =14d

solve for d

d= 55.14

now plug the vALUE OF "d" in one equation (which one you like)

you will get a = 51 , means the first term :)