The seconds term of a geometric series is 4 while the fourth term is 16. Find the sum of the first five terms

Analysis

To solve this problem, we need to use the properties of a Geometric Progression (GP). A GP is defined by its first term ($a$) and its common ratio ($r$). The $n^{th}$ term of a GP is given by the formula: $T_n = ar^{n-1}$

Concept

1. Identify the given terms: • Second term ($T_2$) = $ar^{2-1} = ar = 4$ --- (Equation 1) • Fourth term ($T_4$) = $ar^{4-1} = ar^3 = 16$ --- (Equation 2)

2. Find $r$ and $a$: Divide Equation 2 by Equation 1: $\frac{ar^3}{ar} = \frac{16}{4}$ $r^2 = 4$ $r = \sqrt{4} = 2$ (We take the positive value for this series)

Substitute $r = 2$ into Equation 1: $a(2) = 4$ $a = 2$

3. Calculate the sum of the first five terms ($S_n$): The formula for the sum of the first $n$ terms of a GP is: $S_n = \frac{a(r^n - 1)}{r - 1}$

Solution

Given $a = 2$, $r = 2$, and $n = 5$:

$S_5 = \frac{2(2^5 - 1)}{2 - 1}$ $S_5 = \frac{2(32 - 1)}{1}$ $S_5 = 2 \times 31$ $S_5 = 62$

Conclusion

Comparing our result with the options provided, we see that 62 corresponds to option B.

Exam Tip: When dealing with geometric series in JAMB, if the numbers are small (like the first five terms), you can also list them out manually to verify your answer:

• $T_1 = a = 2$
• $T_2 = ar = 4$
• $T_3 = ar^2 = 8$
• $T_4 = ar^3 = 16$
• $T_5 = ar^4 = 32$ Sum = $2 + 4 + 8 + 16 + 32 = 62$. This is a great way to double-check your work under pressure!