# The sum of 4 numbers that are part of an arithmetic series is 42. If the sum of the first three numbers is 18 what term of the series is the 4th?

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### 1 Answer

The sum of 4 terms of an arithmetic series is 42. And the sum of the first three terms is 18. We have to determine which term of the series is the fourth number.

The nth term of an arithmetic series can be expressed as a +(n-1)*d, where a is the first term and d is the common difference.

Start with the sum of the first three terms which is 18. This gives a + a + d + a + 2d = 18

=> 3a + 3d = 18

=> a + d = 6 ...(1)

The fourth term is 42 - 18 = 24

Let it be the nth term

=> a + (n - 1)d = 24 ...(2)

From (1) and (2)

6 - d + (n - 1)d = 24

=> (n - 2)d = 18

As n and n - 2 are positive integers and d is not equal to 1, n - 2 can be equal to 2, 3, 6, 9.

The fourth number can be term 4, 5, 8 or 11 of the series

To show that all of these are right, lets take n-1 = 11, n = 12

a + 11d = 24

a + d = 6

10d = 18

d = 1.8

a = 4.2

3(a + d) = 18

4.2 + 11*1.8 = 24

Similarly it can be done for the rest.

**The fourth number can be the 4, 5, 8 or 11th term of the series.**