SOLUTION: In a given sequence the first term is 3, the last term is 58, and the sum of all the terms is 366. What is the common difference?

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Question 1191495: In a given sequence the first term is 3, the last term is 58, and the sum of all the terms is 366. What is the common difference?
Answer by math_tutor2020(3817) About Me  (Show Source):
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d = common difference
n = number of terms
a%5B1%5D = first term = 3
a%5Bn%5D = nth term or last term = 58
S%5Bn%5D = sum of the first n terms = 366


S%5Bn%5D+=+%28n%2F2%29%2A%28a%5B1%5D%2Ba%5Bn%5D%29

366+=+%28n%2F2%29%2A%283%2B58%29

366+=+n%2A%281%2F2%29%2A%2861%29

366+=+30.5n

n+=+366%2F%2830.5%29

n+=+12
There are 12 terms.
Phrased another way, the 12th term is 58, ie a%5B12%5D+=+58

Use this to find the value of d below.
a%5Bn%5D+=+a%5B1%5D%2Bd%28n-1%29

a%5B12%5D+=+3%2Bd%2812-1%29

58+=+3%2Bd%2812-1%29

58+=+3%2B11d

11d%2B3+=+58

11d+=+58-3

11d+=+55

d+=+55%2F11

d+=+5
The common difference is 5.
This means we add 5 to each term to get the next term.
The sequence of twelve terms is 3, 8, 13, ..., 53, 58 and these terms add to 366.

Answer: 5