Evaluate each finite series for the specified number of terms. 1+2+4+...;n=5
The series can be expressed as the sum of a geometric series with first term a=1 and common ratio r=2. The formula for the sum of a geometric series is:
S_n = a(1 - r^n)/(1 - r)
Substituting the values a=1, r=2, and n=5, we get:
S_5 = 1(1 - 2^5)/(1 - 2) = 1(-31)/(-1) = 31
Therefore, the sum of the first 5 terms of the series 1+2+4+... is 31.
To evaluate the finite series 1 + 2 + 4 + ... up to the 5th term (n = 5), we can use the formula for the sum of a geometric series.
The given series is a geometric series with a common ratio of 2, as each term is obtained by multiplying the previous term by 2.
The formula for the sum of a geometric series is:
S = a * (1 - r^n) / (1 - r)
where:
S is the sum of the series
a is the first term of the series
r is the common ratio of the series
n is the number of terms
In this case, a = 1, r = 2, and n = 5. Plugging these values into the formula, we get:
S = 1 * (1 - 2^5) / (1 - 2)
Simplifying further:
S = 1 * (1 - 32) / (-1)
S = -31 / -1
S = 31
Therefore, the sum of the series 1 + 2 + 4 + ... up to the 5th term is 31.