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WAEC Further Mathematics 2022 Paper

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Question : 4

Total: 45

Simplify

\frac{9 \cdot 3^{n+1} - 3^{n+2}}{3^{n+1} - 3^{n}}

3
9
27
81

Solution:

Concept:

Factor out the lowest power of

3

and use the laws of exponents to simplify.

Explanation:

First, rewrite each term using powers of

3

.

9 = 3^2

, so

9 \cdot 3^{n+1} = 3^2 \cdot 3^{n+1} = 3^{n+3}

.

Thus the numerator becomes

3^{n+3} - 3^{n+2}

.

Factor out

3^{n+2}

from the numerator:

3^{n+3} - 3^{n+2} = 3^{n+2}(3 - 1) = 2 \cdot 3^{n+2}

.

Now simplify the denominator:

3^{n+1} - 3^n = 3^n(3 - 1) = 2 \cdot 3^n

.

The expression is therefore:

\displaystyle \frac{2 \cdot 3^{n+2}}{2 \cdot 3^n} = 3^{(n+2)-n} = 3^2 = 9

.

Answer:

9

(Option B).

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