4.5 Q-2

Exercise 4.5 — Question 2

Topic: Sum of Geometric Series

This question involves finding the sum of a geometric series using the standard formulas for finite and infinite geometric series.

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Key Formulas

Sum of $n$ terms of a geometric series: $S_n=\frac{a_1(1-r^n)}{1-r},r\ne 1$

Alternatively, using the last term $a_n=a_1{r}^{n-1}$: $S_n=\frac{a_1-r{a}_n}{1-r},r\ne 1$

Sum of an infinite geometric series (converges only when $|r|<1$): $S_{\infty }=\frac{a_1}{1-r}$

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Worked Approach

Step 1: Identify the first term $a_1$ and the common ratio $r=\frac{a_{n+1}}{a_n}$.

Step 2: Determine whether the series is finite (use $S_n$) or infinite (use $S_{\infty }$ if $|r|<1$).

Step 3: Substitute into the appropriate formula and simplify.

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Example

Find the sum of the geometric series: $3+6+12+24+\cdots$ up to 8 terms.

Solution:

• $a_1=3$, $r=\frac{6}{3}=2$, $n=8$
• Since $r\ne 1$: $S_8=\frac{3(2^8-1)}{2-1}=\frac{3(256-1)}{1}=3\times 255=765$

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Example (Infinite Series)

Find the sum of the infinite series: $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots$

Solution:

• $a_1=1$, $r=\frac{1}{2}$, $|r|<1$ ✓ $S_{\infty }=\frac{1}{1-\frac{1}{2}}=\frac{1}{\frac{1}{2}}=2$

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