Sequence – Geometric – One Problem at a Time

Problem 227: Consider the sequence $a = \{2, 10, 50, 250, 1250, \dots \}$ .

a. What is the first term of the sequence?

b. What is the common ration?

c. What is the closed form for the $n^th$ term of the sequence? $a_n$

d. Use the closed form to determine the value of $a_9$ .

Solution:

a. The first term is given by the first number in the sequence. Thus $a_1 = 2$ .

b. One can find the common ratio by dividing consecutive terms. That is,

(1) $\begin{align*} \frac{10}{5} & = 5 \\ \frac{50}{10} & = 5 \\ \frac{250}{5} & = 5.\end{align*}$

Thus the common ratio is 5.

c. Notice that $a_1 = 2 \cdot 5^{1-1} = 2, a_2 = 2 \cdot 5^{2-1} = 10, a_3 = 2 \cdot 5^{3-1} =50, \cdots$ . If you see the pattern, we can write it in general.

(2) $\begin{equation*} a_n = 2 \cdot 5^{n-1}\end{equation*}$

d. Using (2), we can find $a_9$ .

(3) $\begin{align*} a_n &= 2 \cdot 5^{n-1} \\ a_9 & = 2 \cdot 5^{9-1} \\ & = 2 \cdot 5^8 \\ & = 781,250. \end{align*}$

Sequence – Geometric

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