A sequence of numbers $$a_1, a_2, a_3,\ldots$$ is defined as follows: $$a_1=3, a_2=5,$$ and every term in the sequence

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A sequence of numbers $$a_1, a_2, a_3,\ldots$$ is defined as follows: $$a_1=3, a_2=5,$$ and every term in the sequence

by VJesus12 » Fri Jan 28, 2022 7:24 am

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A sequence of numbers $$a_1, a_2, a_3,\ldots$$ is defined as follows: $$a_1=3, a_2=5,$$ and every term in the sequence after $$a_2$$ is the product of all terms in the sequence preceding it, e.g, $$a_3=(a_1)(a_2)$$ and $$a_4=(a1)(a2)(a3).$$ If $$a_n=t$$ and $$n>2,$$ what is the value of $$a_{n+2}$$ in terms of $$t?$$

(A) $$4t$$
(B) $$t^2$$
(C) $$t^3$$
(D) $$t^4$$
(E) $$t^8$$

Source: Official Guide

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Re: A sequence of numbers $$a_1, a_2, a_3,\ldots$$ is defined as follows: $$a_1=3, a_2=5,$$ and every term in the sequen

by [email protected] » Fri Jan 28, 2022 7:37 am
VJesus12 wrote:
Fri Jan 28, 2022 7:24 am
A sequence of numbers $$a_1, a_2, a_3,\ldots$$ is defined as follows: $$a_1=3, a_2=5,$$ and every term in the sequence after $$a_2$$ is the product of all terms in the sequence preceding it, e.g, $$a_3=(a_1)(a_2)$$ and $$a_4=(a1)(a2)(a3).$$ If $$a_n=t$$ and $$n>2,$$ what is the value of $$a_{n+2}$$ in terms of $$t?$$

(A) $$4t$$
(B) $$t^2$$
(C) $$t^3$$
(D) $$t^4$$
(E) $$t^8$$

Source: Official Guide
Let's list a few terms....
term1 = 3
term2 = 5
term3 = (term2)(term1) = (5)(3) = 15 (term2)(term1)
term4 = (term3)(term2)(term1) = (15)(5)(3) = 15²
term5 = (term4)(term3)(term2)(term1) = (15²)(15)(5)(3) = 15⁴
term6 = (term5)(term4)(term3)(term2)(term1) = (15⁴)(15²)(15)(5)(3) = 15⁸

At this point, we can see the pattern.

Continuing, we get....
term7 = 15^16
term8 = 15^32

Each term in the sequence is equal to the SQUARE of term before it

If term_n =t and n > 2, what is the value of term_n+2 in terms of t?
So, term_n = t
term_n+1 = t²
term_n+2 = t⁴