Question

In an AP, the first term is 22,nth term is- 11 and the sum to first n term 66. Find n and d, the common difference​

Answer 1

S O L U T I O N :

$\underline{\bf{Given\::}}$

• First term, (a) = 22
• nth term, (l) = -11
• Sum of nth term, (Sn) = 66

$\underline{\bf{Explanation\::}}$

Using formula of sum of an A.P;

$\boxed{\bf{S_n = \frac{n}{2} [a+l]}}$

A/q

$\mapsto\tt{S_n = \dfrac{n}{2} \times [a+l]}$

$\mapsto\tt{66 = \dfrac{n}{2} \times [22+(-11)]}$

$\mapsto\tt{66 = \dfrac{n}{2} \times [22-11]}$

$\mapsto\tt{66 = \dfrac{n}{2} \times 11}$

$\mapsto\tt{66 \times 2 = 11n}$

$\mapsto\tt{132 = 11n}$

$\mapsto\tt{n = \cancel{132/11}}$

$\mapsto\bf{n = 12}$

Now, using formula of the A.P;

$\boxed{\bf{a_n = a+(n-1)d}}$

$\mapsto\tt{-11 = 22 + (12- 1)d}$

$\mapsto\tt{-11 = 22 + (11)d}$

$\mapsto\tt{-11-22 = 11d}$

$\mapsto\tt{-33= 11d}$

$\mapsto\tt{d= -\cancel{33/11}}$

$\mapsto\bf{d = -3}$

Thus,

The A.P of  n & d will be 12 & -3 .