Question

\frac { 3 ^ { p + 1 } - 3 ^ { p } } { 2 \times 3 ^ { p } } = 1​

Answer 1

True for all p

Explanation:

Given equation:

`(3^(p+1)-3^(p))/(2 * 3^(p))=1`

Multiply both sides by
`2 * 3^p`:

`\implies 3^(p+1)-3^p=2 * 3^p`

Add
`3^p` to both sides:

`\implies 3^(p+1)-3^p+3^p=2 * 3^p+3^p`

`\implies 3^(p+1)=3 * 3^p`

Rewrite 3 as 3¹:

`\implies 3^(p+1)=3^1 * 3^p`

`\textsf{Apply exponent rule} \quad a^b \cdot a^c=a^(b+c)`

`\implies 3^(p+1)=3^(1+p)`

`\implies 3^(p+1)=3^(p+1)`

`\textsf{Apply exponent rule} \quad a^(f(x))=a^(g(x)) \implies f(x)=g(x):`

`\implies p+1=p+1`

Subtract 1 from both sides:

`\implies p+1-1=p+1-1`

`\implies p=p`

Therefore as both sides are equal, true for all p.

Answer 2

Given equation:

• 
  `(3^(p+1) - 3^p) /(2 *3^p) = 1`

Solving for p:

• 
  `3^(p+1) - 3^p = 2 *3^p`
• 
  `3*3^p - 3^p = 2 *3^p`
• 
  `2 *3^p =2 *3^p`

p - any real number

We can state the initial expression is correct for any value of p.