For each geometric series, calculate t6 and S6.
a) 6 + 30 + 150
... So I'm not exactly looking for an answer to this question but I was wondering why they want you to find t6 too? Would finding out t6 (via general formula) help in any way when finding S6? I'm not trying to sass the textbook or anything, I just want to know if I'm missing anything.
Or are they just asking for t6 just to get us to continue practicing general formula?
I think it's just for practice. finding a term and a partial sum are frequent operations, so it's good to know how to get them.
Okay, thanks. Just wanted to make sure I wasn't missing anything!
In a geometric series, each term is obtained by multiplying the preceding term by a constant ratio. The general formula for the nth term (tn) of a geometric series is given by:
tn = a * r^(n-1)
where a is the first term and r is the common ratio.
Now, to answer your question, finding t6 can be helpful when calculating S6 (the sum of the first 6 terms). The sum of the first n terms of a geometric series (Sn) can be calculated using the following formula:
Sn = a * (1 - r^n) / (1 - r)
So, in order to calculate S6, we need the value of t6, which can be found using the general formula for the nth term.
Therefore, in this case, finding t6 is necessary to calculate S6 using the given formulas. It's a way of ensuring you understand the formulas and can apply them correctly.
Calculating t6, the sixth term, and S6, the sum of the first six terms, of a geometric series are related calculations. The general formula for the nth term of a geometric series (tn) is given by:
tn = a * r^(n-1)
where a is the first term of the series and r is the common ratio between consecutive terms.
To calculate t6, you substitute n = 6 into the formula:
t6 = a * r^(6-1)
= a * r^5
Finding t6 can help you find S6, the sum of the first six terms of the series, because the sum of a geometric series (Sn) is given by:
Sn = a * (1 - r^n) / (1 - r)
To calculate S6, you substitute n = 6 into the formula:
S6 = a * (1 - r^6) / (1 - r)
So, finding t6 is important in calculating S6 because you need the value of the sixth term to find the sum of the first six terms.
In this case, the geometric series is given as 6 + 30 + 150. To find t6, you need to identify the first term (a) and the common ratio (r). By examining the series, we can see that the first term is 6 and the common ratio can be found by dividing any term by its preceding term:
r = (30 / 6) = 5
r = (150 / 30) = 5
Now you can calculate t6 and S6 using the formulas mentioned earlier:
t6 = 6 * (5^5) = 6 * 3125 = 18,750
S6 = 6 * (1 - 5^6) / (1 - 5) = 6 * (1 - 15,625) / -4 = 23,430