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# Which term of the AP: $ -2,-7,-12, \ldots $ will be $ -77 $ ? Find the sum of this AP upto the term $ -77 $.

Given:

Given A.P. is $-2, -7, -12, …$

To do:

We have to find which term of the given A.P. is $-77$ and the sum of this A.P. upto the term $-77$.

Solution:

Here,

$a_1=-2, a_2=-7, a_3=-12$

Common difference $d=a_2-a_1=-7-(-2)=-7+2=-5$

We know that,

nth term $a_n=a+(n-1)d$

Therefore,

$a_{n}=-2+(n-1)(-5)$

$-77=-2+n(-5)-1(-5)$

$-77+2=-5n+5$

$75+5=5n$

$5n=80$

$n=\frac{80}{5}$

$n=16$

We know that,Sum of $n$ terms in an A.P. $S_n = \frac{n}{2}[2a + (n – 1) d]$

Therefore, the sum of the given A.P. upto the term $-77$ is,

$S_{16} = \frac{16}{2}[2 (-2) + (16 – 1)(-5)]$

$= 8[-4 + (-75)]$

$= 8(-79)$

$=-632$

Therefore, $-77$ is the 16th term of the given A.P. and the sum of the A.P. upto the term $-77$ is $-632$.

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