$2/5, 2/25, 2/125, 2/625,...$ to n terms.
$1/2(1-(1/5^n))$
Geometric Progression Sum Calculation
The problem asks for the sum of the first n terms of a Geometric Progression (G.P.).
The given G.P. is: $2/5, 2/25, 2/125, 2/625,...$
Identify G.P. Parameters
- First term ($a$): $a = \frac{2}{5}$
- Common ratio ($r$): Calculate by dividing the second term by the first term. $r = \frac{2/25}{2/5} = \frac{2}{25} \times \frac{5}{2} = \frac{5}{25} = \frac{1}{5}$
Since $r = \frac{1}{5}$, which is less than 1, we use the formula for the sum of n terms of a G.P.
Sum of G.P. Formula
The sum ($S_n$) of the first n terms of a G.P. is given by:
$S_n = \frac{a(1 - r^n)}{1 - r}$Calculate the Sum
Substitute the values of $a$ and $r$ into the formula:
$S_n = \frac{\frac{2}{5} \left(1 - \left(\frac{1}{5}\right)^n\right)}{1 - \frac{1}{5}}$Simplify the denominator:
$1 - \frac{1}{5} = \frac{5}{5} - \frac{1}{5} = \frac{4}{5}$Now substitute this back into the sum formula:
$S_n = \frac{\frac{2}{5} \left(1 - \left(\frac{1}{5}\right)^n\right)}{\frac{4}{5}}$Simplify the expression:
$S_n = \frac{2}{5} \times \frac{5}{4} \times \left(1 - \left(\frac{1}{5}\right)^n\right)$ $S_n = \frac{10}{20} \times \left(1 - \left(\frac{1}{5}\right)^n\right)$ $S_n = \frac{1}{2} \left(1 - \left(\frac{1}{5}\right)^n\right)$This can also be written as:
$S_n = \frac{1}{2} \left(1 - \frac{1}{5^n}\right)$Conclusion
The sum of the G.P. to n terms is $\frac{1}{2}(1 - \frac{1}{5^n})$. This matches Option 3.
Important Questions from Geometric Progression
In a school of 750 boys, 34% are Muslims, 42% Hindus, 12% Sikhs and the rest belong to other communities. How many boys are from other community?
Ms. Tina has 2 parents, 4 grand parents, 8 great grand parents and so on. Find the number of her ancestors during 10 (ten) generations preceeding her own.
A square is drawn by joining the midpoints of the sides of a given square. In the same way, this process continues indefinitely. If side of the first square is 4 cm, than the sum of the area of all the squares are equal to
Third term of sequence an = (-1)n(n − 7) is :
811/3 × 811/9 × 811/27 × 811/81 × ______ up to infinite term will be equal to
