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

1 answer

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 \( S_{18} = -234 \)