Chapter 12.1, Problem 75E

nth Term of a Sequence Find a formula for the nth term of the sequence

$\sqrt{2},\sqrt{2\sqrt{2}},\sqrt{2\sqrt{2\sqrt{2}}},\sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}},\dots$

[Hint: Write each term as a power of 2.]

To determine

The $n\text{th}$ term of given sequence.

Answer to Problem 75E

The $n\text{th}$ term of the given expression is $2^{\frac{2^n-1}{2^n}}$.

Explanation of Solution

The given sequence is,

$\sqrt{2},\sqrt{2\sqrt{2}},\sqrt{2\sqrt{2\sqrt{2}}},\sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}}\cdots$ (1)

The sequence follows pattern geometrically to the power of 2,

This shows the above sequence can be rewritten as,

$2^{\frac{1}{2}},2^{\frac{3}{4}},2^{\frac{7}{8}}\cdots$

Observe that numerator has a incremental value and follows a general pattern with the denominator of power 2 which can be generalize as $2^{\frac{2^n-1}{2^n}}$, here n is the nth term of sequence.

Thus, the $n\text{th}$ term of the given expression is $2^{\frac{2^n-1}{2^n}}$.