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mr_godi [17]
3 years ago
9

Drew has 4 pieces of rope 1 meter ong each. He cuts each rope into fifths. How many fifths will he have after cutting all the ro

pes
Mathematics
1 answer:
OleMash [197]3 years ago
5 0
For this question it would make more sense to understand it if you keep each 1 m long piece of rope separate. For every one hole, you would be able to get five groups of 1/5 from it. If there are four of these, you would have four groups of five 1/5. This would be 20 total pieces. Mathematically, you are taking 4 and breaking it into equal groups (division) of 1/5. This will become 4 x 5/1=20.
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The given series is geometric with common ratio 6^x - 9, which converges if |6^x - 9| (i.e. the interval of convergence). We have the well-known result

\displaystyle |r| < 1 \implies \sum_{n=0}^\infty ar^n = \frac{a}{1-r}

If you're not familiar with that result, it's easy to reproduce.

Let S_N be the N-th partial sum of the infinite series,

\displaystyle S_N = \sum_{n=0}^N \left(6^x - 9\right)^n = 1 + \left(6^x - 9\right) + \left(6^x - 9\right)^2 + \cdots + \left(6^x - 9\right)^N

Multiply both sides by the ratio.

\left(6^x - 9\right) S_N = \left(6^x - 9\right) + \left(6^x - 9\right)^2 + \left(6^x - 9\right)^3 + \cdots + \left(6^x - 9\right)^{N+1}

Subtract this from S_N to eliminate all the powers of the ratio between 0 and N+1.

\left(1 - \left(6^x - 9\right)\right) S_N = 1 - \left(6^x - 9\right)^{N+1}

Solve for S_N.

S_N = \dfrac{1 - \left(6^x - 9\right)^{N+1}}{10-6^x}

Now as N\to\infty, the exponential term converges to 0 and we're left with

\displaystyle \sum_{n=0}^\infty \left(6^x-9\right)^n = \lim_{N\to\infty} S_N = \boxed{\frac1{10-6^x}}

7 0
2 years ago
(this is 6 grade math) PLEASE HELP<br>​
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Donavan received 6 votes for every vote that Karen received it donavan recieved 2136 votes how many votes did Karen receive
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