Predicting What Comes Next: Sequences and Progressions | EOT

Question 10

Which term of the GP: $2, 8, 32, \dots$ is $131072$ ? Write the explicit formula as well as the recursive formula for the $n^{\text{th}}$ term.

Solution

Understand the Question

In a Geometric Progression (GP), each subsequent term is obtained by multiplying the preceding term by a constant value called the common ratio ( $r$ ).
For the GP $2, 8, 32, \dots$ , the first term is $a = 2$ and the common ratio is $r = \dfrac{8}{2} = 4$ .
The explicit formula expresses the $n^{\text{th}}$ term directly in terms of $n$ : $a_n = a r^{n-1}$ .
The recursive formula expresses each term using the previous term: $a_1 = 2$ and $a_n = r \cdot a_{n-1}$ for $n \ge 2$ .
To find which term equals $131072$ , we set $a_n = 131072$ and solve for $n$ .

Step 1 · Find the First Term and Common Ratio

Given GP: $2, 8, 32, \dots$

First term ( $a$ ): $a = 2$

Common ratio ( $r$ ): $r = \dfrac{8}{2} = 4$

Step 2 · Write Explicit and Recursive Formulas

Explicit Formula:

\begin{aligned} a_n &= a r^{n-1} \\[0.6em] a_n &= 2 \cdot 4^{n-1} \end{aligned}

Alternatively, in powers of $2$ : $a_n = 2 \cdot (2^2)^{n-1} = 2 \cdot 2^{2n-2} = 2^{2n-1}$

Recursive Formula: $a_1 = 2, \quad a_n = 4a_{n-1} \quad \text{for } n \ge 2$

Step 3 · Find the Term Number $n$ for $131072$

Let the $n^{\text{th}}$ term be $a_n = 131072$ .

Using the explicit formula:

\begin{aligned} 2 \cdot 4^{n-1} &= 131072 \\[0.6em] 4^{n-1} &= \dfrac{131072}{2} \\[0.6em] 4^{n-1} &= 65536 \end{aligned}

Expressing $65536$ as a power of $4$ : $65536 = 4^8$

Equating exponents:

\begin{aligned} n - 1 &= 8 \\[0.6em] n &= 9 \end{aligned}

Thus, $131072$ is the $9^{\text{th}}$ term.

Answer

The $9^{\text{th}}$ term is $131072$ .

Common Mistakes

Index Exponent Error: Using $a_n = a r^n$ instead of $a_n = a r^{n-1}$ , which shifts all term indices by one.
Incomplete Recursive Definition: Writing only $a_n = 4a_{n-1}$ while omitting the initial condition ( $a_1 = 2$ ) and the domain constraint ( $n \ge 2$ ).
Base Conversion Mistakes: Mixing powers of $2$ and $4$ without converting both sides to the same base.

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