document.write( "Question 1094327: How many terms from the sequence 5, 1, 0.2, 0.04, .....are needed to form a sum that is within 2.5 x 10^ -8 of the infinite sum? \n" ); document.write( "

Algebra.Com's Answer #708948 by greenestamps(13200) $\"\"$ $\"About$

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The infinite sum is
\n" ); document.write( " $\"5%2F%281-1%2F5%29+=+5%2F%284%2F5%29+=+25%2F4+=+6.25\"$

\n" ); document.write( "Now look at the sequence consisting of the infinite sum minus the sum of the first n terms of the given sequence. The sums of the first n terms of the given sequence form the sequence
\n" ); document.write( "5, 6, 6.2, 6.24, ...

\n" ); document.write( "The sequence consisting of the differences between the infinite sum and the terms in this sequence is
\n" ); document.write( "1.25, .25, .05, .01, ...

\n" ); document.write( "We can see that this sum is a decreasing geometric sequence, with first term 1.25 and common ratio 0.2. We want to know how many terms we need to go out in this sequence to get a number less than 2.5x10^-8. So we need to solve
\n" ); document.write( " $\"1.25%280.2%29%5E%28n-1%29+%3C+2.5%2A10%5E-8\"$

\n" ); document.write( "Logarithms and a scientific calculator show that 12 terms is just barely not enough; we need to add 13 terms of the given sequence to get within 2.5*10^-8 of the infinite sum.
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