Official Solution -- Two detailed ways to solve:


Test nos. :
To get a handle on this tough geometric sequence problem, try sketching an example and testing the choices. An example sequence is given in the problem. Another example with simpler numbers starts with the number 5 and uses 2 as the constant multiplier. The sequence would then be:

5, 10, 20, 40, 80

Now test the possible answer choices:

A: YES. 3(5), 3(10), 3(20), 3(40), 3(80) = 15, 30, 60, 120, 240. Notice that each term is still increasing by the constant multiple of 2. This is a geometric sequence. One example does not guarantee that the sequence will always be geometric, but if you changed the numbers, the multiple would remain the same. The only thing different would be the starting number (3 times the old starting number). The result is always a geometric sequence.

B: NO. 2 + 2, 4 + 2, 8 + 2, 16 + 2, 32 + 2, 64 + 2 = 4, 6, 10, 18, 34, 66. Test the first few terms in the sequence. To move from 4 to 6, one must multiply by . However to move from 6 to 10, one must multiply by . Because these multipliers are not the same, this cannot be a geometric sequence. Eliminate this choice.

C: YES. \(5^2, 10^2, 20^2, 40^2, 80^2\) = 20, 100, 400, 1,600, 6,400. To move from 25 to 100, multiply by 4. To move from 100 to 400, again multiply by 4. The same holds for all following terms, so this is a geometric sequence. Again, a single example does not prove that the result will always be geometric, but notice that the original constant multiple (2) changed into a new constant multiple (4) in a predictable way (22 = 4). This pattern always holds true, and the result is always a geometric sequence.

D: YES.
\(√5, √10, √20, √40, √80 = √5, √10, 2√5, 2√10, 4√5.\) This sequence is toughest to evaluate, but observe that to get from one term to the next, you have to multiply by a constant multiple of \(√2\) . Therefore, this is a geometric sequence. Once more, a single example doesn’t prove a general case, but you can notice the predictable change to the constant multiple: the original value of 2 is square-rooted. This pattern always holds true, and the result is always a geometric sequence.


Algebraic way. :
An algebraic proof involves rewriting the original sequence in terms of a (the starting number) and k, making up a new letter for the constant multiplier. The original sequence a, b, c, d, e can then be rewritten as \(a, ak, ak^2, ak^3, ak^4\).

Next, you rewrite each answer choice the same way and see whether the new sequence is itself in the form of a new starting point (a new “a”) and a new constant multiplier (a new “k”). For instance, answer choice A becomes this:

\(3a, 3ak, 3ak^2, 3ak^3, 3ak^4\)

The new starting point is 3a, while the constant multiplier is still just k. In contrast, answer choice B becomes this:

\(a + 2, ak + 2, ak^2 + 2, ak^3 + 2, ak^4 + 2\)

There is no way to analyze this sequence as having a constant multiplier from term to term.

Answer choice C becomes this:

\(a^2, (ak)^2, (ak^2)^2, (ak^3)^2, (ak^4)^2\)

You can now apply exponent rules:\( a^2, a^2k^2, a^2k^4, a^2k^6, a^2k^8\). The new starting point is \(a^2\), while the new constant multiplier is \(k^2\), as you can see by examining each successive term. This is still a geometric sequence.

Finally, a similar analysis on answer choice D shows you that the new starting point is \(√a\), while the new constant multiplier is \(√k\). Again, the sequence remains geometric.

The correct answers are A, C, and D.