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# Geometric sequence

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Find the fifth term of the geometric sequence with first term 2 and second term 1/5.

Jul 3, 2021

#1
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so we have  $\{ \underbrace{2}_\text{ first term },\underbrace{\frac{1}{5}}_\text{ 2nd term } , \underbrace{...}_\text{ 2 other terms }, \underbrace{a_5}_\text{ 5th term } \}$

the formula to find any term of this geosequence is

$a_n=ar^{n-1}$ where $a_n=n^{th} term$    ;   $a=\text{first term}$   ;   $r=\text{common ratio}$

so far we have $a_5=2r^{5-1}$

to find the ratio we just write this simple equation $2\cdot \:r=\frac{1}{5} \implies r=\frac{1}{10}$

now that we know what the common ratio is, going back and plugging it in the formula we get:

$a_5=2\left(\frac{1}{10}\right)^{5-1}$

$a_5=2\cdot \frac{1^4}{10^4}$

$a_5=2\cdot \frac{1}{10^4}$

$a_5=\frac{2}{10^4}$

$\boxed{a_5=\frac{2}{10000}} \Leftrightarrow \boxed{ a_5=\frac{1}{5000}}$

Jul 3, 2021
#2
+33606
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From a1 to a2   is    a1 * r     so r = 1/10

starting form second term   1/5 * r * r * r   = a5 = 1/5 * 1/1000 = 1/5000

Jul 3, 2021