Question

A sequence \( a_{1}, a_{2}, a_{3}, \ldots \) is defined as follows: \( a_{1}=3 \), and \( a_{k}=4 a_{k-1}+2 \) for all integers \( k \geq 2 \). (a) Find \( a_{1}, a_{2} \), and \( a_{3} \). (b) Supposing that \( a_{5}=4^{4} \cdot 3+4^{3} \cdot 2+4^{2} \cdot 2+4 \cdot 2+2 \), find a similar numerical expression for \( a_{6} \) by substituting the right-hand side of this equation in place of \( a_{5} \) in the equation \[ a_{6}=4 \cdot a_{5}+2 \] (c) Guess an explicit formula for \( a_{n} \). Simplify your answer using one of the following reference formulas: \[ 1+2+3+\cdots+n=[n(n+1)] / 2 \text { for all integers } n \geq 1 \] \( 1+r+r^{2}+\cdots+r^{m}=\left(r^{m+1}-1\right) /(r-1) \) for all integers \( m \geq 0 \) and all real numbers \( r \neq 1 \).

          A sequence \( a_{1}, a_{2}, a_{3}, \ldots \) is defined as follows:
\( a_{1}=3 \), and \( a_{k}=4 a_{k-1}+2 \) for all integers \( k \geq 2 \).
(a) Find \( a_{1}, a_{2} \), and \( a_{3} \).
(b) Supposing that \( a_{5}=4^{4} \cdot 3+4^{3} \cdot 2+4^{2} \cdot 2+4 \cdot 2+2 \), find a similar numerical expression for \( a_{6} \) by substituting the right-hand side of this equation in place of \( a_{5} \) in the equation
\[
a_{6}=4 \cdot a_{5}+2
\]
(c) Guess an explicit formula for \( a_{n} \). Simplify your answer using one of the following reference formulas:
\[
1+2+3+\cdots+n=[n(n+1)] / 2 \text { for all integers } n \geq 1
\]
\( 1+r+r^{2}+\cdots+r^{m}=\left(r^{m+1}-1\right) /(r-1) \) for all integers \( m \geq 0 \) and all real numbers \( r \neq 1 \).

A sequence a1, a2, a3, … is defined as follows:
a1=3, and ak=4 ak-1+2 for all integers k ≥ 2.
(a) Find a1, a2, and a3.
(b) Supposing that a5=4^4· 3+4^3· 2+4^2· 2+4 · 2+2, find a similar numerical expression for a6 by substituting the right-hand side of this equation in place of a5 in the equation

a6=4 · a5+2

(c) Guess an explicit formula for an. Simplify your answer using one of the following reference formulas:

1+2+3+⋯+n=[n(n+1)] / 2 for all integers n ≥ 1

1+r+r^2+⋯+r^m=(r^m+1-1) /(r-1) for all integers m ≥ 0 and all real numbers r ≠ 1.

Added by Alex J.

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