How many terms of the $A.P.$ $-6,-\frac{11}{2},-5, \ldots \ldots$ are needed to give the sum $-25 ?$
How many terms of the $A.P.$ $-6,-\frac{11}{2},-5, \ldots \ldots$ are needed to give the sum $-25 ?$
Let the sum of $n$ terms of the given $A.P.$ be $-25$
It is known that,
$S_{n}=\frac{n}{2}[2 a+(n-1) d]$
Where $n=$ number of terms, $a=$ first term, and $d=$ common difference
Here, $a=-6$
$d=-\frac{11}{2}+6=\frac{-11+12}{2}=\frac{1}{2}$
Therefore, we obtain
$-25=\frac{n}{2}\left[2 \times(-6)+(n-1)\left(\frac{1}{2}\right)\right]$
$\Rightarrow-50=n\left[-12+\frac{n}{2}-\frac{1}{2}\right]$
$\Rightarrow-50=n\left[-\frac{25}{2}+\frac{n}{2}\right]$
$\Rightarrow-100=n(-25+n)$
$\Rightarrow n^{2}-25 n+100=0$
$\Rightarrow n^{2}-5 n-20 n+100=0$
$\Rightarrow n(n-5)-20(n-5)=0$
$\Rightarrow n=20$ or $5$
Similar Questions
If the numbers $a,\;b,\;c,\;d,\;e$ form an $A.P.$, then the value of $a - 4b + 6c - 4d + e$ is
If the numbers $a,\;b,\;c,\;d,\;e$ form an $A.P.$, then the value of $a - 4b + 6c - 4d + e$ is
If the angles of a quadrilateral are in $A.P.$ whose common difference is ${10^o}$, then the angles of the quadrilateral are
If the angles of a quadrilateral are in $A.P.$ whose common difference is ${10^o}$, then the angles of the quadrilateral are
Insert five numbers between $8$ and $26$ such that resulting sequence is an $A.P.$
Insert five numbers between $8$ and $26$ such that resulting sequence is an $A.P.$
A man deposited $Rs$ $10000$ in a bank at the rate of $5 \%$ simple interest annually. Find the amount in $15^{\text {th }}$ year since he deposited the amount and also calculate the total amount after $20$ years.
A man deposited $Rs$ $10000$ in a bank at the rate of $5 \%$ simple interest annually. Find the amount in $15^{\text {th }}$ year since he deposited the amount and also calculate the total amount after $20$ years.
If the sum of a certain number of terms of the $A.P.$ $25,22,19, \ldots \ldots .$ is $116$ Find the last term
If the sum of a certain number of terms of the $A.P.$ $25,22,19, \ldots \ldots .$ is $116$ Find the last term