The sum till infinity of a Geometric Progression, with first term 16 and common ratio $$r$$, is 48. Find the sum till infinity of another GP with first term 76 and a common ratio of $$\left(r-\dfrac{1}{6}\right)$$.

Given that the first term of GP is 16 and sum till infinity is 24

So, $$S_∞=\dfrac{a}{1-r}$$, where $$a$$ is the first term and $$r$$ is the common ratio.

$$S_∞=\dfrac{16}{1-r}=48$$

$$1-r=\dfrac{16}{48}=\dfrac{1}{3}$$

or, $$r=\dfrac{2}{3}$$

Now, we have to find sum till infinity of another GP whose first term is 76 and common ratio is $$r_1=\left(r-\dfrac{1}{6}\right)=\dfrac{2}{3}-\dfrac{1}{6}=\dfrac{1}{2}$$

$$S_∞=\dfrac{76}{1-\dfrac{1}{2}}$$

$$S_∞=152$$

Hence, the answer is 152