Question

The first and last term of an ap are 21 and -47 if the sum of the series is given as - 234 calculate the number of terms in the ap, the common difference and the sum of the first 18 terms

Answer 1

Let's denote the first term of the arithmetic progression (AP) as \( a_1 = 21 \) and the last term (or \( n \)-th term) as \( a_n = -47 \).

The sum of the first \( n \) terms of an AP can be calculated with the formula:

\[
S_n = \frac{n}{2} (a_1 + a_n)
\]

According to the problem, the sum of the series is given as \( S_n = -234 \). Substituting the values we know:

\[
-234 = \frac{n}{2} (21 + (-47))
\]

Calculating \( 21 + (-47) \):

\[
21 - 47 = -26
\]

Now substituting back into the equation:

\[
-234 = \frac{n}{2} (-26)
\]

Multiplying both sides by \( -2 \):

\[
468 = n \cdot 26
\]

Now, divide by 26:

\[
n = \frac{468}{26} = 18
\]

So, the number of terms \( n = 18 \).

Next, we need to find the common difference \( d \). The formula for the \( n \)-th term of an AP is given by:

\[
a_n = a_1 + (n-1) d
\]

Substituting the values we have:

\[
-47 = 21 + (18 - 1) d
\]

This simplifies to:

\[
-47 = 21 + 17d
\]

Now, isolate \( d \):

\[
-47 - 21 = 17d
\]
\[
-68 = 17d
\]
\[
d = \frac{-68}{17} = -4
\]

Thus, the common difference \( d = -4 \).

Finally, we need to calculate the sum of the first 18 terms, which we have already calculated as \( S_{18} = -234 \).

To summarize:
- The number of terms \( n = 18 \)
- The common difference \( d = -4 \)
- The sum of the first 18 terms \( S_{18} = -234 \)