Answer:
970/3.
Explanation:
If you are bored of arithmetic series, then you might want to try geometric series. They are much more fun and exciting, because they involve multiplying by a constant ratio instead of adding a constant difference. For example, the series 1 + 2 + 4 + 8 + ... is geometric, because each term is twice the previous one. That means you can get really big numbers really fast, which is always cool.
But how do you find the sum of a geometric series? Well, there is a formula for that, and it's not too hard to remember. It goes like this:
S_n = a_1 (1 - r^n) / (1 - r), where a_1 is the first term, r is the common ratio and n is the number of terms.
That's it! Just plug in the values and you're done. But wait, there's more! Sometimes, you might not know all the values, and you have to do some algebra to find them. For example, what if you are given that S_n = 728, r = 3 and the last term a_n = 486? How do you find n and a_1?
Don't panic, it's not as hard as it looks. You just have to use another formula for the n^th term of a geometric series:
a_n = a_1 * r^(n-1)
Then you can solve for n and a_1 using some logarithms and some basic equations. Here are the steps:
First, we can find n by using the formula for the n^th term of a geometric series:
a_n = a_1 * r^(n-1)
Substituting the given values, we get:
486 = a_1 * 3^(n-1)
Dividing both sides by a_1, we get:
486 / a_1 = 3^(n-1)
Taking the logarithm of both sides with base 3, we get:
log_3 (486 / a_1) = n - 1
Adding 1 to both sides, we get:
n = log_3 (486 / a_1) + 1
Next, we can find a_1 by using the formula for the sum of a geometric series:
S_n = a_1 (1 - r^n) / (1 - r)
Substituting the given values and the value of n we found, we get:
728 = a_1 (1 - 3^(log_3 (486 / a_1) + 1)) / (1 - 3)
Simplifying, we get:
728 = a_1 (486 / a_1 - 3) / (-2)
Multiplying both sides by (-2), we get:
-1456 = a_1 (486 / a_1 - 3)
Expanding, we get:
-1456 = 486 - 3 * a_1
Adding 3 * a_1 to both sides, we get:
-970 = -3 * a_1
Dividing both sides by -3, we get:
a_1 = 970 / 3
Therefore, the first term of the series is 970/3.
And that's how you do it! Easy peasy lemon squeezy! Now you can impress your friends and teachers with your geometric series skills. Just don't forget to check your answers and show your work. Have fun!