Expert Answer to "The following series are neither arithmetic nor geometric,..."

abreviatsjw

Answered question

2022-01-13

The following series are neither arithmetic nor geometric, but by analyzing their patterns, you can find their sums. Find the sum of each series.

\sum_{n = 1}^{12} (2^{n} - 1)

Ben Owens

Beginner2022-01-14Added 27 answers

Step 1
We need to find the sum of the series,

\sum_{n = 1}^{12} (2^{n} - 1)

Step 2
We have

\sum_{n = 1}^{12} (2^{n} - 1) = (\sum_{n = 1}^{12} 2^{n}) - (\sum_{n = 1}^{12} 1)

Now,

\sum_{n = 1}^{12} 1 = \underset{12 times}{\underset{⏟}{1 + 1 + 1 + \dots + 1}} = 12

Step 3
And the series

\sum_{n = 1}^{12} 2^{n}

is geometric, with first term

t_{1} = 2^{1} = 2

common ratio

r = 2

, and total number of terms

n = 12

. Hence,

\sum_{n = 1}^{12} 2^{n} = \frac{t_{1} (r^{n} - 1)}{r - 1} = \frac{2 (2^{12} - 1)}{2 - 1} = 2 (2^{12} - 1) = 8190

Step 4
Thus we have the sum of the given series,

\sum_{n = 1}^{12} (2^{n} - 1) (\sum_{n = 1}^{12} 2^{n}) - (\sum_{n = 1}^{12} 1)

= 8190 - 12

= 8178

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