Question

which of the AP 85,78,71,... is its first negative term? if the prder of this term is n, find Sn.

Answer 1

85, 78, 71....
d = 78-85 = -7
a = 85
Let the first negative number of the AP be x
so x < 0
a + (n-1)d < 0
85 + (n-1)-7 < 0
-7n + 7 < -85
-7n < -92
n > 92/7
n > 13.14

So the 14ᵗʰ number will be negaitve
which is 85 + (14-1)-7 = 85 + 13(-7) = 85 - 91 = -6


Sn=
$\frac{n}{2} (2a + (n - 1)d$
$\frac{n}{2} (2(85) + (n - 1)( - 7))$

Answer 2

Given

a=85

d=-7

We know that the nth term is a+(n-1)d

Let nth term be the first negative term.

a+(n-1)d<0

85+(n-1)-7<0

85-7n+7<0

7n>92

The least possible value of n is 14

for which 7*14=98.

Term=a+(n-1)d

=a+13d

=85-91

=-6

S=n/2*[2a+(n-1)d]

=14/2*[170+13*-7]

=7*[170-91]

=7*79

=553

That will be the sum of the terms till the first negative term.

Guess this is what you meant in the question.

If you want to find the sum of any n term:

then S=n/2*[2a+(n-1)(-7)]

S=n/2*[170-7n+7]

Anyways the sum till the first negative is 553 and there are 14 terms till the first negative,-6 will be the first negative term.

Hope it helps you.