Answer:
True for all p
Step-by-step explanation:
Given equation:
[tex]\dfrac{3^{p+1}-3^{p}}{2 \times 3^{p}}=1[/tex]
Multiply both sides by [tex]2 \times 3^p[/tex]:
[tex]\implies 3^{p+1}-3^p=2 \times 3^p[/tex]
Add [tex]3^p[/tex] to both sides:
[tex]\implies 3^{p+1}-3^p+3^p=2 \times 3^p+3^p[/tex]
[tex]\implies 3^{p+1}=3 \times 3^p[/tex]
Rewrite 3 as 3¹:
[tex]\implies 3^{p+1}=3^1 \times 3^p[/tex]
[tex]\textsf{Apply exponent rule} \quad a^b \cdot a^c=a^{b+c}[/tex]
[tex]\implies 3^{p+1}=3^{1+p}[/tex]
[tex]\implies 3^{p+1}=3^{p+1}[/tex]
[tex]\textsf{Apply exponent rule} \quad a^{f(x)}=a^{g(x)} \implies f(x)=g(x):[/tex]
[tex]\implies p+1=p+1[/tex]
Subtract 1 from both sides:
[tex]\implies p+1-1=p+1-1[/tex]
[tex]\implies p=p[/tex]
Therefore as both sides are equal, true for all p.
