Question

The sum of six terms of an ap is 9 and the sum if 9 terms is 6 find the sum of fifteen terms

Answer 1

Answer:  The sum of first fifteen terms of the A.P. is -15.

Step-by-step explanation:  Given that the sum of first six terms of an A.P. is 9 and the sum of first nine terms of the same A.P. is 6.

We are to find the sum of first fifteen terms.

Let a and d represents the first term and the common difference of the given A.P.

Then, the sum of first n terms will be given by

$S_n=\dfrac{n}{2}(2a+(n-1)d).$

According to the given information, we have

$S_6=9\\\\\\\Rightarrow \dfrac{6}{2}(2a+(6-1)d)=9\\\\\\\Rightarrow 3(2a+5d)=9\\\\\Rightarrow 2a+5d=3~~~~~~~~~~~~~~~~~~~~~~~~~(i)$

and

$S_9=6\\\\\\\Rightarrow \dfrac{9}{2}(2a+(9-1)d)=6\\\\\\\Rightarrow 2a+8d=\dfrac{4}{3}~~~~~~~~~~~~~~~~~~~~~~~~~(ii)$

Subtracting equation (i) from equation (ii), we get

$(2a+8d)-(2a+5d)=\dfrac{4}{3}-3\\\\\\\Rightarrow 3d=\dfrac{4-9}{3}\\\\\\\Rightarrow d=-\dfrac{5}{9}.$

From equation (i), we get

$2a+5\times(-\dfrac{5}{9})=3\\\\\\\Rightarrow 2a-\dfrac{25}{9}=3\\\\\\\Rightarrow 2a=3+\dfrac{25}{9}\\\\\\\Rightarrow 2a=\dfrac{52}{9}\\\\\\\Rightarrow a=\dfrac{26}{9}.$

Therefore, the sum of first fifteen terms will be

$S_{15}\\\\\\=\dfrac{15}{2}(2a+(15-1)d)\\\\\\=\dfrac{15}{2}\left(2\times\dfrac{26}{9}+14\times(-\dfrac{5}{9})\right)\\\\\\=\dfrac{15}{2}\left(\dfrac{52-70}{9}\right)\\\\\\=\dfrac{15}{2}\times(-\dfrac{18}{9})\\\\=-15.$

Thus, the sum of first fifteen terms of the A.P. is -15.

Answer 2

Answer:

-15 is the answer............