Question

Evaluate \[ \left(\frac{4}{7}+\frac{1}{3}\right)+\left(\frac{4}{7}+\frac{4}{3}\right)\left(\frac{1}{3}\right) +\left(\frac{4}{7}+\frac{4}{3}\right)^2\left(\frac{1}{3}\right)^2+\cdots \]

• A. $\dfrac{5}{2}$
• B. $5$
• C. $\dfrac{7}{2}$
• D. $\dfrac{8}{3}$

Hint:

Always rewrite long series in standard G.P. form before applying formulas.

Answer 1

The Correct Option is A

Step 1: Identify the series.
The given series is a geometric progression with \[ a=\frac{4}{7}+\frac{1}{3},\quad r=\frac{1}{3} \] Step 2: Simplify first term.
\[ a=\frac{12+7}{21}=\frac{19}{21} \] Step 3: Apply infinite G.P. formula.
\[ S=\frac{a}{1-r} \] \[ S=\frac{\frac{19}{21}}{1-\frac{1}{3}}=\frac{\frac{19}{21}}{\frac{2}{3}} \] \[ S=\frac{19}{14}=\frac{5}{2} \] Final conclusion.
The value of the given expression is $\dfrac{5}{2}$.