Question
In an $AP: a_n= 4, d = 2, S_n= -14,$ find $n$ and $a.$
✓
Answer
Here, $a_n= 4$
$d = 2$
$S_n= -14$
We know that
$a_n= a + (n - 1)d$
$ \Rightarrow 4 = a + (n - 1)d$
$ \Rightarrow 4 = a + 2n - 2$
$ \Rightarrow 4 + 2 = a + 2n$
$ \Rightarrow 6 = a + 2n$
$ \Rightarrow a + 2n = 6 ...... (1)$
Again, we know that
${S_n} = \frac{n}{2}\left[ {2a + (n - 1)d} \right]$
$ \Rightarrow - 14 = \frac{n}{2}\left[ {2a + (n - 1)2} \right]$
$ \Rightarrow -14 = n[a + (n - 1)]$
$ \Rightarrow -14 = n (a + n - 1)$
$ \Rightarrow -14 = n (6 - n - 1) ....... $From $(1), (a + 2n = 6 \Rightarrow a + n = 6 - n)$
$ \Rightarrow -14 = n(-n + 5)$
$ \Rightarrow -14 = -n^2+ 5n$
$ \Rightarrow n^2- 7n + 2n - 14 = 0$
$ \Rightarrow n(n - 7) + 2(n - 7) = 0$
$ \Rightarrow (n - 7) (n + 2) = 0$
$ \Rightarrow n - 7 = 0$ or $n + 2 = 0$
$ \Rightarrow n = 7 $ or $n = -2$
$ \Rightarrow n = - 2$ is in admissible as $n,$ being the number of terms, is a natural number.
$\therefore n = 7$
Putting $n = 7$ in equation $(1),$ we get
$a + 2(7) = 6$
$ \Rightarrow a + 14 = 6$
$ \Rightarrow a = 6 - 14$
$ \Rightarrow a = -8 $
$d = 2$
$S_n= -14$
We know that
$a_n= a + (n - 1)d$
$ \Rightarrow 4 = a + (n - 1)d$
$ \Rightarrow 4 = a + 2n - 2$
$ \Rightarrow 4 + 2 = a + 2n$
$ \Rightarrow 6 = a + 2n$
$ \Rightarrow a + 2n = 6 ...... (1)$
Again, we know that
${S_n} = \frac{n}{2}\left[ {2a + (n - 1)d} \right]$
$ \Rightarrow - 14 = \frac{n}{2}\left[ {2a + (n - 1)2} \right]$
$ \Rightarrow -14 = n[a + (n - 1)]$
$ \Rightarrow -14 = n (a + n - 1)$
$ \Rightarrow -14 = n (6 - n - 1) ....... $From $(1), (a + 2n = 6 \Rightarrow a + n = 6 - n)$
$ \Rightarrow -14 = n(-n + 5)$
$ \Rightarrow -14 = -n^2+ 5n$
$ \Rightarrow n^2- 7n + 2n - 14 = 0$
$ \Rightarrow n(n - 7) + 2(n - 7) = 0$
$ \Rightarrow (n - 7) (n + 2) = 0$
$ \Rightarrow n - 7 = 0$ or $n + 2 = 0$
$ \Rightarrow n = 7 $ or $n = -2$
$ \Rightarrow n = - 2$ is in admissible as $n,$ being the number of terms, is a natural number.
$\therefore n = 7$
Putting $n = 7$ in equation $(1),$ we get
$a + 2(7) = 6$
$ \Rightarrow a + 14 = 6$
$ \Rightarrow a = 6 - 14$
$ \Rightarrow a = -8 $
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