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Find the sum of the geometric series 176 + 88 + 44 + ⋅⋅⋅ + 11.
Find the sum of the geometric series 176 plus 88 plus 44 plus ... plus 11.
The sum of the first 𝑛 terms of a geometric series is given by 𝑎 multiplied by one minus 𝑟 to the power of 𝑛 divided by one minus 𝑟, where 𝑎 is the first term of the series, 𝑟 is the common ratio, and 𝑛 is the number of terms in the series.
The first term in our series is 176. Therefore, 𝑎 is equal to 176. The second term of our series is half of the first term. The third term is half of the second term, and so on. Therefore, to get from the first term to the second, we need to multiply by 0.5. This means that our common ratio 𝑟 is 0.5.
Our next step is to calculate 𝑛 — the number of terms in the series. How many terms are there between 44 and 11? Well, 44 multiplied by 0.5 is 22 and 22 multiplied by 0.5 is 11. Therefore, there was only one missing term — 22. This means that our value for 𝑛 is five. There are five terms in the series.
Substituting in our values of 𝑎, 𝑟, and 𝑛 into the formula gives us 176 multiplied by one minus 0.5 to the power of five divided by one minus 0.5. Typing this into the calculator gives us an answer of 341.
This means that the sum of the geometric series with first term 176, common ratio 0.5, and five terms is equal to 341.
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