A Test of Reasoning, Not Math

by | Brain Teasers

Below we once again share an official GMAT question as our monthly brain teaser. Like many difficult GMAT and GRE questions, and even like many of the most difficult SAT/ACT questions, this one shares a lot in common with brain teasers. The question looks very difficult, even unapproachable, but the solution is quite easy and requires only basic math ability. As is often the case, it is the reasoning that one needs to apply that is the key!

    \[ \text{If } x \neq 1 \text{ then } \frac{x^n - 1}{x - 1} = x^{n-1} + x^{n-2} + \dots + x^2 + x + 1. \] \[ \text{Given that } 1 + 7 + 7^2 + \dots + 7^8 = 6{,}725{,}601,\ \text{what does } 7^9 \text{ equal?} \] \textbf{A. }\, 6(6{,}725{,}601) + 1 \\ \textbf{B. }\, 6(6{,}725{,}601 + 1) \\ \textbf{C. }\, 8(6{,}725{,}601) + 1 \\ \textbf{D. }\, (6{,}725{,}601 - 1)(6 + 1) \\ \textbf{E. }\, (6{,}725{,}601)(6 + 1)

Answer:
Though this is a GMAT Quant question, we’re posting this as a “brain teaser” because the kind of thinking that one needs to apply to get this question right is very similar to that which one needs to apply on brain teasers! And, typical of GMAT questions, the math is ultimately very easy. It’s the problem solving and reasoning that one needs to apply that is difficult. That’s the GMAT for you!
Most test takers just don’t know where to get started on this question. It looks crazy at first sight! But (and this is a similarity with brain teasers), you need to think about what you’re being given and what the question is asking for. Upon looking more carefully, it should be clear that the “formula” given (with the x’s and n’s) looks a lot like the 1 + 7^2… = 6,725,601 that follows. There is clearly some connection between those two equations!
Ok, but still, what to do? This is where trying to be more “tangible” or “concrete” can be helpful (especially in the face of things that are abstract or confusing), and this is a MAJOR strategy that we teach here at Reason Test Prep. Well, the question is asking for the value of 7^9. Perhaps we could put a 7 into the given formula as our value of x, since that is the base in the formula and the base of what the question is asking for? And then perhaps we could put in an exponent for n? Well, probably the best exponent to insert would be 9, since that is what the question is asking for, but let’s imagine we chose 8, since that is the highest power in the given equation (the 1 + 7^2 +…+ 7^8 = 6,725,601 equation). We would get:

    \[ \frac{7^8 - 1}{7 - 1} = 7^7 + 7^6 + \cdots + 7^2 + 7 + 1 \]

Ok, that’s interesting. That almost matches the second equation they gave us! If we just raised that power by 1 and inserted n=9, we’d get something on the right side of the equation that exactly matches what we were given:

    \[ \frac{7^9 - 1}{7 - 1} = 7^8 + 7^7 + \cdots + 7^2 + 7 + 1 \]

Well, we know that the right side of the above equation equals 6,725,601, since that’s what we were given. Great! We can now just replace the right side of the above equation with 6,725,601!

    \[ \frac{7^9 - 1}{7 - 1} = 6{,}725{,}601 \]

From here, it’s just basic algebra. We just need to solve for 7^9:

    \[ \frac{7^9 - 1}{6} = 6{,}725{,}601 \] \[ 7^9 - 1 = 6(6{,}725{,}601) \] \[ 7^9 = 6(6{,}725{,}601) + 1 \]

So the correct answer is A. The beauty of this question, like so many GMAT questions and so many quantitative-style brain teasers alike, is that the math itself is very easy. This is probably not beyond 7th grade math: you don’t even need to understand exponents to get this question right, just basic algebra. However, the reasoning and problem solving ability that one needs to get this question right do make this question very difficult. Remember, when solving GMAT, GRE, or even difficult SAT or ACT questions, just like when solving brain teasers, you must put your thinking cap on and look to apply robust and dynamic reasoning and critical thinking. Aspire to be a clever problem-solver, not a robotic Math-doer!