Question

Find the 81st term of the arithmetic sequence minus, 10, comma, minus, 25, comma, minus, 40, comma, dot dot dot−10,−25,−40,...

Answer 1

To find the 81st term of the arithmetic sequence given by \(-10, -25, -40, \ldots\), we first identify the properties of the sequence.

The first term \(a\) is:
\[
a = -10
\]

Next, we find the common difference \(d\) of the sequence. The common difference is calculated by subtracting the first term from the second term:
\[
d = -25 - (-10) = -25 + 10 = -15
\]

The general formula for the \(n\)-th term \(a_n\) of an arithmetic sequence is:
\[
a_n = a + (n-1)d
\]

We need to find the 81st term (\(n = 81\)):
\[
a_{81} = a + (81 - 1)d
\]
This simplifies to:
\[
a_{81} = a + 80d
\]

Substituting the values of \(a\) and \(d\):
\[
a_{81} = -10 + 80(-15)
\]

Now, calculate \(80 \times -15\):
\[
80 \times -15 = -1200
\]

So now we can substitute back:
\[
a_{81} = -10 - 1200 = -1210
\]

Thus, the 81st term of the arithmetic sequence is:
\[
\boxed{-1210}
\]