document.write( "Question 583415: Can you help me figure out what the formula is in this pattern: 8,24,72,216 ? Example of formula:3n-3 N=the position of the number \n" ); document.write( "

Algebra.Com's Answer #372637 by KMST(5328) $\"\"$ $\"About$

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There are two special sequence patterns that are studies in algebra:
\n" ); document.write( "arithmetic sequences (called arithmetic progressions in other countries) and
\n" ); document.write( "geometric sequences (called geometric progressions in other countries).
\n" ); document.write( "In an arithmetic sequence each number (we call them terms) is the one before plus a constant that we call the common difference. To find out if a sequence is an arithmetic sequence, we calculate the differences between consecutive terms. In this case, we calculate:
\n" ); document.write( " $\"24-8=16\"$ ,
\n" ); document.write( " $\"72-24=48\"$ , and that is enough to realize that the differences (16 and 48) are not the same. There is no common difference. It is not an arithmetic sequence.
\n" ); document.write( "In a geometric sequence, each term is equal to the previous number times a constant that we call the common ratio. To find out if a sequence is a geometric sequence, we calculate the ratios of consecutive terms. In this case, we calculate:
\n" ); document.write( " $\"24%2F8=3\"$ ,
\n" ); document.write( " $\"72%2F24=3\"$ , and
\n" ); document.write( " $\"216%2F72=3\"$ .
\n" ); document.write( "The ratio is always 3. There is a common ratio and it's 3. It is a geometric sequence. We call the nth term $\"a%5Bn%5D\"$ , and we know that, calling the common ratio $\"r\"$ ,
\n" ); document.write( " $\"a%5Bn%5D=a%5Bn-1%5D%2Ar\"$ or $\"a%5Bn%2B1%5D=a%5Bn%5D%2Ar\"$
\n" ); document.write( "That's what we call a recursive formula.
\n" ); document.write( "We also know that, calling the first tern $\"a%5B1%5D\"$ ,
\n" ); document.write( " $\"a%5Bn%5D=a%5B1%5D%2Ar%5E%28n-1%29\"$ , because
\n" ); document.write( " $\"a%5B2%5D=a%5B1%5D%2Ar\"$ ,
\n" ); document.write( " $\"a%5B3%5D=a%5B2%5D%2Ar=a%5B1%5D%2Ar%2Ar=a%5B1%5D%2Ar%5E2\"$ ,
\n" ); document.write( " $\"a%5B4%5D=a%5B3%5D%2Ar=a%5B1%5D%2Ar%5E2%2Ar=a%5B1%5D%2Ar%5E3\"$ , and so on.
\n" ); document.write( "We can do the formal proof by induction, if needed, but $\"a%5Bn%5D=a%5B1%5D%2Ar%5E%28n-1%29\"$ is a very useful formula to calculate the nth term.
\n" ); document.write( "In your case,
\n" ); document.write( " $\"a%5B1%5D=8\"$ , $\"r=8\"$ and
\n" ); document.write( " $\"highlight%28a%5Bn%5D=8%2A3%5E%28n-1%29%29\"$ \n" ); document.write( "

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