The sum of a geometric series is 126. The first term and the common ratio are both 2. How many terms are in the series? A. 5 B. 6 C. 7 D. 8
7 years ago
Answered By Maximilian W
For a geometric series, we know that the sum of the first n terms can be expressed by:
sn = a1+a2+a3+...+an = a1 x [(1-rn)/(1-r)] , where sn is the sum of the first n terms (126), a1 is the first term in the series (2), and r is the common ratio (2). We are looking for n, which is the number of terms in the series that will give us our answer.
Since we want to find n, we have to rearrange the formula to isolate n. For now, let's multiply both sides of the equation by the (1-r) term and divide both sides by the a1 term. Doing this, we get:
[(sn x (1-r)) / a1] = 1-rn
Now, we add rn to both sides and subtract [(sn x (1-r)) / a1] from both sides to get:
rn = 1-[(sn x (1-r)) / a1]
If we plug in all of our values, we get:
2n = 1 - [(126 x (1-2))/2] = 1 - [-126 / 2] = 1 - [-63] = 1+63 = 64 ---> 2n = 64, and n=the number of terms in the series (our answer).
Now, you can guess and check to see that 2n=64 means that n=6 (2x2x2x2x2x2=64). OR, if you have learned about logarithms, you can solve the problem like this:
Step One: log both sides --> log(2n) = log(64)
Step Two: Since the n is an exponent, it can be brought in front of the log() as a multiplier --> n x log(2) = log(64)
Step Three: Divide both sides by log(2) to get: n = log(2)/log(64). Plugging that into the calculator should give n=6. Hope this helps!
7 years ago
Answered By Maximilian W
For a geometric series, we know that the sum of the first n terms can be expressed by:
sn = a1+a2+a3+...+an = a1 x [(1-rn)/(1-r)] , where sn is the sum of the first n terms (126), a1 is the first term in the series (2), and r is the common ratio (2). We are looking for n, which is the number of terms in the series that will give us our answer.
Since we want to find n, we have to rearrange the formula to isolate n. For now, let's multiply both sides of the equation by the (1-r) term and divide both sides by the a1 term. Doing this, we get:
[(sn x (1-r)) / a1] = 1-rn
Now, we add rn to both sides and subtract [(sn x (1-r)) / a1] from both sides to get:
rn = 1-[(sn x (1-r)) / a1]
If we plug in all of our values, we get:
2n = 1 - [(126 x (1-2))/2] = 1 - [-126 / 2] = 1 - [-63] = 1+63 = 64 ---> 2n = 64, and n=the number of terms in the series (our answer).
Now, you can guess and check to see that 2n=64 means that n=6 (2x2x2x2x2x2=64). OR, if you have learned about logarithms, you can solve the problem like this:
Step One: log both sides --> log(2n) = log(64)
Step Two: Since the n is an exponent, it can be brought in front of the log() as a multiplier --> n x log(2) = log(64)
Step Three: Divide both sides by log(2) to get: n = log(2)/log(64). Plugging that into the calculator should give n=6. Hope this helps!