The price of a certain commodity increased at a rate of

Hi there! I'm happy to help with this!

Let's say, the price in 2000 is A. That the original amount. Each year it increases X%. To represent a percent increase (a) write the percent as a fraction/decimal, here X/100; (b) add one ---> 1 + X/100; (c) that's the multiplier -- multiplying a number by that multiplier results in a X% increase.

price in 2000 = A
price in 2001 = A*(1 + X/100)
price in 2002 = A*(1 + X/100)^2
price in 2003 = A*(1 + X/100)^3
price in 2004 = A*(1 + X/100)^4

For simplicity, I am going to define r = (1 + X/100). Then these equations become:

price in 2000 = A
price in 2001 = A*r
price in 2002 = A*r^2
price in 2003 = A*r^3
price in 2004 = A*r^4

Now, suppose we have M = 2001 price = A*r and N = 2003 price = A*r^3. How do we represent the 2002 prince (A*r^2) in terms of M and N?

There are two methods.

Method One: express r in terms of M and N

This is more a crank-it-out algebraic solution approach. We notice that N/M = (A*r^3)/(A*r) = r^2, so r = sqrt(N/M). Well,

2002 price = (2001 price)*(r) = M * sqrt(N/M) = [sqrt(M)*sqrt(M)]*[sqrt(N)/sqrt(M)] = sqrt(M)*sqrt(N) = sqrt(NM).

Through some fast-and-loose manipulation of the laws of squareroots, we arrive at answer .

Method Two: a more elegant solution for a more civilized age . . .

When you have an arithmetic sequence --- that is, adding the same number to get new terms (e.g. 8, 11, 14, 17, 20, 23, . . . ), when you take any three numbers in a row, the middle number is the mean, the arithmetic average, of the outer two. For example ---11, 14, 17 --- (11 + 17)/2 = 14. The arithmetic average is the ordinary average --- add the two numbers, and divide by two.

When you have a geometric sequence -- that is, multiplying the same ratio to get new terms (e.g. 2, 6, 18, 54, 162, . . . ), when you take any three numbers in a row, the middle number is the geometric mean of the outer two. The geometric mean of two numbers means multiply the two numbers and take the squareroot. For example --- 6, 18, 54 --- 6*54 = 324, and sqrt(324) = 18.

When you apply a fixed percentage increase from one term to the next, as we have in this problem, that's a geometric sequence. Thus, to find the 2002 price, all you have to do is take the geometric mean of the 2001 price and the 2003 price. 2002 price = sqrt(MN). Bam. Done. Again, answer = .

The ideas about arithmetic & geometric sequences, and the associated means, are good tricks to have up your sleeve for the more challenging GMAT math problems.

Does all this make sense? Please let me know if you have any questions.

Mike