The sum of the first 6 terms of a geometric series with a common ratio of 2 is given by 126. Calculate the 5th term of the series
a(2^6 - 1)/(2-1) = 126
a=2
a_5 = 2*2^4 = 32
To find the 5th term of the series, we first need to determine the value of the first term of the series.
In a geometric series, the sum of the first n terms can be expressed as:
Sn = a(1 - r^n) / (1 - r),
where:
Sn is the sum of the first n terms,
a is the first term of the series,
r is the common ratio of the series, and
n is the number of terms.
In this case, we are given that the sum of the first 6 terms is 126, so we have:
126 = a(1 - 2^6) / (1 - 2).
Simplifying the equation:
126 = a(1 - 64) / (-1).
Multiplying both sides by -1:
-126 = a(63).
Dividing both sides by 63:
a = -126 / 63 = -2.
Now that we know the first term, we can calculate the 5th term of the series by using the formula:
Tn = a * r^(n-1),
where Tn is the nth term of the series.
Substituting the given values:
T5 = -2 * 2^(5-1).
Simplifying the equation:
T5 = -2 * 2^4 = -2 * 16 = -32.
Therefore, the 5th term of the geometric series is -32.