Question
Find the sum of the G.P.:
$5/11, 5/121, 5/1331, 5/14641, ...$ to $n$ terms.
The correct answer is
$1/2(1-(1/11)^n)$
Finding the Sum of the Geometric Progression
The given series is a Geometric Progression (G.P.): $5/11, 5/121, 5/1331, 5/14641, \dots$
Identify G.P. Parameters
- The first term, $a$, is $5/11$.
- To find the common ratio, $r$, divide the second term by the first term: $r = \frac{5/121}{5/11} = \frac{5}{121} \times \frac{11}{5} = \frac{11}{121} = \frac{1}{11}$
- The number of terms is $n$.
Apply the Sum Formula
The formula for the sum of the first $n$ terms of a G.P. where $|r| < 1$ is:
$ 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{5}{11} \left(1 - \left(\frac{1}{11}\right)^n\right)}{1 - \frac{1}{11}} $First, calculate the denominator:
$ 1 - \frac{1}{11} = \frac{11}{11} - \frac{1}{11} = \frac{10}{11} $Now substitute this back into the sum formula:
$ S_n = \frac{\frac{5}{11} \left(1 - \left(\frac{1}{11}\right)^n\right)}{\frac{10}{11}} $Simplify the expression:
$ S_n = \frac{5}{11} \times \frac{11}{10} \times \left(1 - \left(\frac{1}{11}\right)^n\right) $ $ S_n = \frac{5}{10} \times \left(1 - \left(\frac{1}{11}\right)^n\right) $ $ S_n = \frac{1}{2} \left(1 - \left(\frac{1}{11}\right)^n\right) $Conclusion
The sum of the G.P. to $n$ terms is $\frac{1}{2}(1-(\frac{1}{11})^n)$.
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Important Questions from Progression (Notes)
- The sum of 16 terms of the series $\sqrt{2} + \sqrt{8} + \sqrt{18} + \sqrt{32} + .....$ is :
- The $5^{\text{th}}$ and $9^{\text{th}}$ terms of an arithmetic progression are 7 and 13 respectively. What is the $15^{\text{th}}$ term?
The value of $1 + (\frac{1}{2^1} + \frac{1}{3}) + (\frac{1}{2^2} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7}) + ... + (\frac{1}{2^9} + ... + \frac{1}{1023})$ lies between
- A ball is dropped from a height of 100 m. The ball after each bounce rises vertically by half its previous height (This means at the first bounce it rises by 50 m, by 25 m at the second bounce and so on). What is the vertical distance travelled by the ball between the first and the fifth bounces?
- An ant starts at the origin and moves along the $y$-axis and covers a distance $l$. This is its first stage in its journey. Every subsequent stage requires the ant to turn right and move a distance which is half of its previous stage. What would be its coordinates at the end of its $5^{\text{th}}$ stage?
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