Question

The 10th and 15th terms of an Ap are -5 and -7 1/2, respectively, what is the sum of the first 20 terms

Answer 1

To find the sum of the first 20 terms of the arithmetic progression (AP) given that the 10th term (\(a_{10}\)) is -5 and the 15th term (\(a_{15}\)) is -7.5, we start by using the general formula for the \(n\)th term of an AP:

\[
a_n = a + (n - 1)d
\]

where \(a\) is the first term and \(d\) is the common difference.

From the information provided:

1. The 10th term:
\[
a_{10} = a + 9d = -5
\]

2. The 15th term:
\[
a_{15} = a + 14d = -7.5
\]

Now we can set up the two equations based on these terms:

\[
(1) \quad a + 9d = -5
\]
\[
(2) \quad a + 14d = -7.5
\]

Next, we can subtract equation (1) from equation (2) to eliminate \(a\):

\[
(a + 14d) - (a + 9d) = -7.5 - (-5)
\]

This simplifies to:

\[
5d = -2.5
\]

Now, solve for \(d\):

\[
d = \frac{-2.5}{5} = -0.5
\]

Now that we have \(d\), we can substitute \(d\) back into equation (1) to find \(a\):

\[
a + 9(-0.5) = -5
\]
\[
a - 4.5 = -5
\]
\[
a = -5 + 4.5 = -0.5
\]

Now we have both \(a\) (the first term) and \(d\) (the common difference):
- \(a = -0.5\)
- \(d = -0.5\)

Next, we can find the sum of the first 20 terms of the AP. The formula for the sum of the first \(n\) terms \(S_n\) of an arithmetic sequence is given by:

\[
S_n = \frac{n}{2} \cdot (2a + (n - 1)d)
\]

Setting \(n = 20\):

\[
S_{20} = \frac{20}{2} \cdot (2(-0.5) + (20 - 1)(-0.5))
\]
\[
= 10 \cdot (2(-0.5) + 19(-0.5))
\]
\[
= 10 \cdot (-1 + (-9.5))
\]
\[
= 10 \cdot (-10.5)
\]
\[
= -105
\]

Therefore, the sum of the first 20 terms of the arithmetic progression is

\(\boxed{-105}\).