How can I find the 27th term of an arithmetic sequence where the first term is 38 and the 17th term is 74?

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To find the 27th term of an arithmetic sequence, we first need to understand how the sequence works. An arithmetic sequence is defined by its first term and a common difference between consecutive terms.

In this case, we are given:

  • First term (
    a1
    ) = 38
  • 17th term (
    a17
    ) = 74

To determine the common difference (
d
) of the sequence, we can use the formula for the nth term of an arithmetic sequence, which is:

an = a1 + (n - 1)d

Plugging in the values for the 17th term:

a17 = a1 + (17 - 1)d

This can be rearranged to:

a17 = 38 + 16d

We know that
a17
= 74, so:

74 = 38 + 16d

Now, solving for
d
: 

74 - 38 = 16d
36 = 16d
d = 36 / 16
d = 2.25

Now that we have the common difference, we can find the 27th term
(a27)
using the formula again:

a27 = a1 + (27 - 1)d
= 38 + 26(2.25)
= 38 + 58.5
= 96.5

Therefore, the 27th term of the arithmetic sequence is:

96.5

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