Caleb Snyder

2022-01-23

Consider the sequences
a. $\left\{{a}_{n}\right\}=\left\{-2,5,12,19,\dots \right\}$
b. $\left\{{b}_{n}\right\}=\left\{3,6,12,24,48,\dots \right\}$
(a) Find the next two terms of the sequence.
(b) Find a recurrence relation that generates the sequence.
(c) Find an explicit formula for the nth term of the sequence.

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1)Given sequence is an arythmetic sequence
First term ${a}_{1}=-2$
Common difference $d=5-\left(-2\right)=5+2=7$
Next two terms are
${a}_{5}={a}_{1}+\left(5-1\right)d=-2+\left(4\right)7=-2+28$
${a}_{5}=26$
${a}_{6}={a}_{1}+\left(6-1\right)d=-2+\left(5\right)7=-2+35$
${a}_{5}=33$
Recurrence formula:
${a}_{1}=-2$
${a}_{n}={a}_{n-1}+7$
Explicit formula
${a}_{5}={a}_{1}+\left(n-1\right)d$
2)Given sequence is a multiple of 2
First term ${a}_{1}=3$
Next two terms:
${a}_{6}=2{a}_{5}=2\left(48\right)=96$
${a}_{7}=2{a}_{6}=2\left(96\right)=192$
Recurrence formula
${a}_{1}=3$
${a}_{n}=2{a}_{n-1}$
Explicit formula
${a}_{n}=3{\left(2\right)}^{n-1}$

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