Question

Write the five first terms of the sequence, whose $n^{th}$ term is $a_n=(-1{)}^{n-1}{5}^{n+1}$.

Answer 1

Finding sequence
Given, $a_n=(-1{)}^{n-1}{5}^{n+1},n\in N$
Substituting $n=1,2,3,4$ and $5$ in $a_n$, we get

$a_1=(-1{)}^{1-1}\cdot 5^{1+1}$
$\Rightarrow a_1=(-1{)}^0\cdot 5^2$
$\Rightarrow a_1=1\times 25$
$\Rightarrow a_1=25$

$a_2=(-1{)}^{2-1}\cdot 5^{2+1}$
$\Rightarrow a_2=(-1{)}^1\cdot 5^3$
$\Rightarrow a_2=(-1)\times 125$
$\Rightarrow a_2=-125$

$a_3=(-1{)}^{3-1}\cdot 5^{3+1}$
$\Rightarrow a_3=(-1{)}^2\cdot 5^4$
$\Rightarrow a_3=1\times 625$
$\Rightarrow a_3=625$

$a_4=(-1{)}^{4-1}\cdot 5^{4+1}$
$\Rightarrow a_4=(-1{)}^3\cdot 5^5$
$\Rightarrow a_4=(-1)\times \left(3125\right)$
$\Rightarrow a_4=-3125$

$a_5=(-1{)}^{5-1}\cdot 5^{5+1}$
$\Rightarrow a_5=(-1{)}^4{.5}^6$
$\Rightarrow a_5=1\times 15625$
$\Rightarrow a_5=15625$

Hence, the first five terms of the sequence are $25,-125,625,-3125,15625$.