Answer:
[tex]\dfrac{5^{n+2}-6\times 5^{n+1}}{13 \times5^{n}-2\times5^{n+1}} = -\dfrac{5}{3}[/tex]
Step-by-step explanation:
We are given the expression to be simplified:
[tex]\dfrac{5^{n+2}-6\times 5^{n+1}}{13 \times5^{n}-2\times5^{n+1}}[/tex]
Let us take common a term with a power of 5 from the numerator and the denominator of the given expression.
We know that:
[tex]a^{p+q} = a^p \times a^q[/tex]
Let us use it to solve the powers of 5 in the given expression.
[tex]\therefore[/tex] we can write:
[tex]5^{n+2} = 5^{n+1}\times 5= 5^n\times 5^{2}[/tex]
[tex]5^{n+1} = 5^n\times 5[/tex]
The given expression becomes:
[tex]\dfrac{5^{n+1} \times 5-6\times 5^{n+1}}{13 \times5^{n}-2\times5^{n}\times 5}[/tex]
Taking common [tex]5^{n+1}[/tex] from the numerator and
Taking common [tex]5^{n}[/tex] from the denominator
[tex]\Rightarrow \dfrac{5^{n+1} (5-6)} {5^{n}(13-2\times5)}\\\Rightarrow \dfrac{5^{n+1} (-1)} {5^{n}(13-10)}\\\Rightarrow -\dfrac{5^{n+1}} {5^{n}\times3}\\\Rightarrow -\dfrac{5^{n}\times 5} {5^{n}\times3}\\\Rightarrow -\dfrac{5}{3}[/tex]
[tex]\therefore[/tex] The answer is:
[tex]\dfrac{5^{n+2}-6\times 5^{n+1}}{13 \times5^{n}-2\times5^{n+1}} = -\dfrac{5}{3}[/tex]
