Which term of the GP: 2, 8, 32, ... is 131072? Write the explicit for…

Question 19 of 51intermediate🔧 ApplyNumerical4 marks

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

Correct Answer

9th term, Explicit formula: $t_n = 2 \times 4^{n-1}$ , Recursive formula: $t_1 = 2, t_n = 4t_{n-1}$ for $n \geq 2$

Exercise: END-OF-CHAPTER EXERCISES | Q: 10 | (Chapter: 22)

For More Understanding

Explanation

To find the term number, we first identify the first term and the common ratio of the given Geometric Progression. Using the explicit formula for the nth term of a GP, we set up an equation and solve for n. The explicit and recursive formulas are then derived using the standard forms provided in the context.

Solution Steps

Identify a and r: For the GP 2, 8, 32, ..., the first term $a = 2$ and the common ratio $r = 8 / 2 = 4$ .
Set up the equation: Let the $n^{th}$ term be 131072. Using the explicit formula $t_n = a \times r^{(n-1)}$ , we get $2 \times 4^{n-1} = 131072$ .
Solve for n: Divide both sides by 2 to get $4^{n-1} = 65536$ . Since $4^8 = 65536$ , we have $n - 1 = 8$ , which gives $n = 9$ . Thus, the 9th term is 131072.
Write Explicit Formula: The explicit formula for the $n^{th}$ term is $t_n = 2 \times 4^{n-1}$ .
Write Recursive Formula: The recursive formula relates each term to the previous term by multiplying by the common ratio. Thus, $t_1 = 2$ and $t_n = 4t_{n-1}$ for $n \geq 2$ .

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