Question

In an arithmetic sequence \{a_n\}, a_1=3, a_4=12. In a sequence \{b_n\}, b_1=4, b_4=20, and \{b_n-a_n\} is a geometric sequence.

What are the formulas for general terms of the sequence \{a_n\} and \{b_n\}?

What is formula for the sum of sequence \{b_n\}?

Collected in the board: Arithmetic sequence

Steven Zheng posted 2 months ago


Answer

Let d as the common difference of the arithmetic sequence \{a_n\}

Using the formula for general terms of an arithmetic sequence,

a_4=a_1+(4-1)d

d=(a_4-a_)/3=(12-3)/3=3

Therefore,

a_n=a_1+(n-1)d

a_n=3+3n-3=3n

Let q as the common ratio of the geometric sequence \{b_n-a_n\}

q^3=\dfrac{b_4-a_4}{b_1-a_1} =\dfrac{20-12}{4-3}=8

Therefore,

q=2

b_n-a_n=(b_1-a_1)q^{n-1}

b_n = 3n+2^{n-1}

Steven Zheng posted 2 months ago

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