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Question
In an A.P. the first term is 8, nth term is 33 and the sum to first n terms is 123. Find n and d, the common differences.
Solution
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.
Here,
The first term of the A.P (a) = 8
The nth term of the A.P (l) = 33
Sum of all the terms Sn = 123
Let the common difference of the A.P. be d.
So, let us first find the number of the terms (n) using the formula,
`123 = (n/2)( 8 + 33)`
`123 = (n/2) (41) `
`((123)(2))/41 = n`
`n = 246/41`
n = 6
Now, to find the common difference of the A.P. we use the following formula,
l = a + (n-1)d
We get,
33 = 8 + (6-1) d
33 = 8 + (5)d
`(33-8)/5 = d`
Further, solving for d,
`d =25/5`
d = 5
Therefore, the number of terms is n = 6 and the common difference of the A.P. d= 5 .
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