Geometric Partial Sum Problem, 3 ways to solve.

Problem:

Find the sum of the first 7 terms of 6 + 18 + 54 + ...

Regardless of method used (except for the method of brute force) we need to figure out that:
{a_n} = 6, 18, 54, etc.
a_n=6(3)^n-1

 

Without a Formulas:

 

or just S₇= 6 + 18 + 54 + ... +4374

or just 3*S₇ = 18 + 54 + ... + 13122  (Multiply your partial sum by your r value, in this case 3.)

3*S₇ - S₇ = 6(3)⁷ - 6(3)⁰  or just use 3*S₇ - S₇= 13122 -6 (Subtract your original series from your result in the prior step.)

Combine like terms: 2*S₇ = 13116  and divide by 2 to get S₇ = 6558.

With a Formula

Note that this is the same answer that you would get if you used the formula:
.

With a Calculator

It is also the same answer that you get if you use your calculator:
    sum(seq(6*3^(X-1),X,1,7)) = 6558
sum is found in the LIST>MATH menu and seq is found in LIST>OPS on the TI-83/84 calculators.

Why know 3 different methods to accomplish the same task?

It is always good to understand a non-calculator method of solving things because you could be given a problem where one of the items that is usually a constant turns out to be a variable or you could simply be asked to show an algebraic procedure for solving a problem. Okay, you might say, but why not just use the formula? Well, if the setup of the problem doesn't exactly match the setup that the formula was derived from, then you need to be able to either use your Properties of Summation skills to manipulate the given expression or just be able to solve without the formula to begin with. Either way, you need strong algebraic skills. Take a look at the following problems for examples of problems that don't always match the formula.

1. For which one can we just use the formula for a finite geometric series without needing to adjust it?
2. How should we adjust the other two to use the formula? Show how to solve the problem with adjusted formulas. Of course you may want to use the method of no formula as described above.

Answers:
A. You can plug directly into the formula for problem number 3.
B.

Last Note

You may be thinking, for all of these problems, I could have just used brute force by calculating each term and then doing the addition. Well it is always nice to explain things and think about things with problems that can be verified with a brute force technique, but don't expect that every problem you are going to run into will be that easy. Some will have so many terms, that it might take you the rest of your natural life to solve it without algebraic techniques. Do take the time to learn these skills.