document.write( "Question 1121118: Let {an}∞n=1 be a sequence whose partial sums are {Sn}∞n=1. Suppose that a1=2 and an=4⋅an−1.\r
\n" ); document.write( "\n" ); document.write( "Find a general formula for the nth term of the sequence of partial sums. \n" ); document.write( "

Algebra.Com's Answer #736893 by greenestamps(13200) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "I will ignore the first sentence of your post, since the notation is not anything I have seen....

\n" ); document.write( "The question you ask can still be answered.

\n" ); document.write( "The sequence has first term 2, and each subsequent term is 4 times the preceding term. So the sequence is

\n" ); document.write( "2, 8, 32, 128, 512, ...

\n" ); document.write( "The partial sums are

\n" ); document.write( "2, 10, 42, 170, 682, ...

\n" ); document.write( "The first partial sum can be written as

\n" ); document.write( "2 = 2(1) = 2(4^0)

\n" ); document.write( "The second partial sum can be written as

\n" ); document.write( "10 = 2(1+4) = 2(4^0+4^1)

\n" ); document.write( "The third partial sum can be written as

\n" ); document.write( "42 = 2(1+4+16) = 2(4^0+4^1+4^2)

\n" ); document.write( "The fourth partial sum can be written as

\n" ); document.write( "170 = 2(1+4+16+64) = 2(4^0+4^1+4^2+4^3)

\n" ); document.write( "The n-th partial sum can be written as

\n" ); document.write( "2(4^0+4^1+4^2+...+4^(n-1))

\n" ); document.write( "The expression in parentheses is a geometric series, for which there is a nice closed form for the sum.

\n" ); document.write( "The general formula for the n-th partial sum of the given sequence is

\n" ); document.write( " $\"2%28%284%5En-1%29%2F%284-1%29%29+=+2%28%284%5En-1%29%2F3%29\"$ \n" ); document.write( "

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