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

Question

GPT-4o mini · 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:

$- 47 = 21 + 17 d$

Now, isolate $d$ :

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

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 $S_{18} = - 234$