In an A.P., the first term is 22, nth term is −11 and the sum to first n terms is 66. Find n and d, the common difference
In the given problem, we have the first and the nth term of an A.P. along with the sum of the n terms of A.P. Here, we need to find the number of terms and the common difference of the A.P.
The first term of the A.P (a) = 22
The nth term of the A.P (l) = −11
Sum of all the terms Sn = 66
Let the common difference of the A.P. be d.
So, let us first find the number of the terms (n) using the formula,
`66 = (n/2) [ 22 + (-11)]`
`66 = (n/2 ) (22 - 11)`
( 66)(2) = (n)(11)
Further, solving for n
`n =( (66)(2))/11`
n = (6) (2)
n = 12
Now, to find the common difference of the A.P. we use the following formula,
l = a + ( n-1) d
-11 = 22 + ( 12 - 1) d
- 11 = 22 + ( 11) d
`(-11 - 22) /11 = d`
Further, solving for d,
`d =( -33)/11`
d = -3
Therefore, the number of terms is n = 12 and the common difference of the A.P.d = -3 .