$5/11, 5/121, 5/1331, 5/14641, ...$ to $n$ terms.
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 $\frac{1}{2}(1-(\frac{1}{11})^n)$.
Important Questions from Progression (Notes)
- The sum of 16 terms of the series $\sqrt{2} + \sqrt{8} + \sqrt{18} + \sqrt{32} + .....$ is :
- A pilgrim starts walking for a journey of 115 km. On the first day he covers 7 km, the next day 9 km, and likewise keeps adding 2 km everyday till he reaches 15 km per day which he maintains for the rest of the journey. How many days in all will he take to complete the journey?
If a, b and c are in Geometric Progression and $a^\frac{1}{x} = b^\frac{1}{y} = c^\frac{1}{z}$ then, x, y, z are in ________.
Which of the following statement is true about the geometric series
$ 1 + r +r^2 + r^3 + ...............; (r > 0) $?
$6240$ रुपये की राशि $30$ किस्तों में इस प्रकार चुकाई जाती है कि प्रत्येक किस्त पिछली किस्त से $10$ रुपये अधिक है । पहली किस्त की मूल्य ____________है।
