$\frac{3(4^5+4^6+4^7+4^8+4^9+4^{10})}{(2^5+2^6+2^7+2^8+2^9+2^{10})} = ?$

Answer Choices

Correct answer marked below

2^{10} + 2^5

2^{10} + 2^6

2^{11} + 2^5

Correct

2^{11} + 2^6

2^{16}

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Explanation

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Official Explanation

Official

We can solve this using the formula for the sum of a geometric progression: Sum = a(r^n - 1)/(r - 1), where a is the first term, r is the common ratio, and n is the number of terms.

For the numerator:

4^5+4^6+4^7+4^8+4^9+4^{10} = 4^5(4^6 - 1)/(4-1) = 4^5(4^6 - 1)/3

For the denominator:

2^5+2^6+2^7+2^8+2^9+2^{10} = 2^5(2^6 - 1)/(2-1)

The original expression becomes:

\frac{3 \times 4^5(4^6 - 1)/3}{2^5(2^6 - 1)} = \frac{4^5(4^6 - 1)}{2^5(2^6 - 1)}

…

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What this question tests

What it tests

Your understanding of divisibility rules, least common multiples, and greatest common factors.

Common trap

Forgetting that 1 is not prime and that every integer divides 0.

3(45+46+47+48+49+410)(25+26+27+28+29+210)=?\frac{3(4^5+4^6+4^7+4^8+4^9+4^{10})}{(2^5+2^6+2^7+2^8+2^9+2^{10})} = ?(25+26+27+28+29+210)3(45+46+47+48+49+410)​=?