G.P. Sum Calculation

This question requires finding the sum of the first n terms of a given Geometric Progression (G.P.).

Identify G.P. Parameters

The provided Geometric Progression is: $2/7, 2/49, 2/343, 2/2401,...$

• First term ($a$): The first term in the sequence is $a = 2/7$.
• Common ratio ($r$): This is found by dividing any term by its preceding term. $r = \frac{2/49}{2/7} = \frac{2}{49} \times \frac{7}{2} = \frac{1}{7}$

Apply G.P. Sum Formula

The standard formula for the sum of the first $n$ terms ($S_n$) of a G.P. is:

$S_n = \frac{a(1 - r^n)}{1 - r}$

Substitute the identified values of $a = 2/7$ and $r = 1/7$ into the formula:

$S_n = \frac{2/7 (1 - (1/7)^n)}{1 - 1/7}$

Simplify the Expression

First, calculate the value of the denominator:

$1 - 1/7 = 6/7$

Now, substitute this value back into the sum formula:

$S_n = \frac{2/7 (1 - (1/7)^n)}{6/7}$

Simplify the coefficient part of the expression:

$\frac{2/7}{6/7} = \frac{2}{7} \times \frac{7}{6} = \frac{2}{6} = \frac{1}{3}$

Therefore, the sum simplifies to:

$S_n = \frac{1}{3} (1 - (1/7)^n)$

Conclusion

The sum of the Geometric Progression $2/7, 2/49, 2/343,...$ up to $n$ terms is $\frac{1}{3} (1 - (1/7)^n)$.