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arithmetic sequence

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A recursive formula can produce arithmetic, geometric or other sequences.

For example, for n = 1, 2, 3, ...:

u0 = 2, un = un-1 + 5 is an arithmetic sequence.

u0 = 2, un = un-1 * 5 is a geometric sequence.

u0 = 0, un = un-1 + n is the sequence of triangular numbers.

u0 = 0, un = un-1 + n(n+1)/2 is the sequence of perfect squares.

u0 = 1, u1 = 1, un+1 = un-1 + un is the Fibonacci sequence.

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Q: Can a recursive formula produce an arithmetic or geometric sequence?

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-7

Yes, it can both arithmetic and geometric.The formula for an arithmetic sequence is: a(n)=a(1)+d(n-1)The formula for a geometric sequence is: a(n)=a(1)*r^(n-1)Now, when d is zero and r is one, a sequence is both geometric and arithmetic. This is because it becomes a(n)=a(1)1 =a(1). Note that a(n) is often written anIt can easily observed that this makes the sequence a constant.Example:a(1)=a(2)=(i) for i= 3,4,5...if a(1)=3 then for a geometric sequence a(n)=3+0(n-1)=3,3,3,3,3,3,3and the geometric sequence a(n)=3r0 =3 also so the sequence is 3,3,3,3...In fact, we could do this for any constant sequence such as 1,1,1,1,1,1,1...or e,e,e,e,e,e,e,e...In general, let k be a constant, the sequence an =a1 (r)1 (n-1)(0) with a1 =kis the constant sequence k, k, k,... and is both geometric and arithmetic.

In this case, 22 would have the value of 11.

They differ in formula.

12, 6, 0, -6, ...

Related questions

what is the recursive formula for this geometric sequence?

4, -1236, -108 is not a geometric system.

-7

Yes, it can both arithmetic and geometric.The formula for an arithmetic sequence is: a(n)=a(1)+d(n-1)The formula for a geometric sequence is: a(n)=a(1)*r^(n-1)Now, when d is zero and r is one, a sequence is both geometric and arithmetic. This is because it becomes a(n)=a(1)1 =a(1). Note that a(n) is often written anIt can easily observed that this makes the sequence a constant.Example:a(1)=a(2)=(i) for i= 3,4,5...if a(1)=3 then for a geometric sequence a(n)=3+0(n-1)=3,3,3,3,3,3,3and the geometric sequence a(n)=3r0 =3 also so the sequence is 3,3,3,3...In fact, we could do this for any constant sequence such as 1,1,1,1,1,1,1...or e,e,e,e,e,e,e,e...In general, let k be a constant, the sequence an =a1 (r)1 (n-1)(0) with a1 =kis the constant sequence k, k, k,... and is both geometric and arithmetic.

In this case, 22 would have the value of 11.

They differ in formula.

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There are different answers depending upon whether the sequence is an arithmetic progression, a geometric progression, or some other sequence. For example, the sequence 4/1 - 4/3 + 4/5 - 4/7 adds to pi

by the general formula ,a+(n-1)*d * * * * * That assumes that it is an arithmetic sequence. The sequence cound by geometric ( t(n) = a*rn ) or power ( t(n) = n2 ) or something else.

I expect that you mean the formula for calculating the terms in a geometric sequence. Please see the link.

You didn't say the series (I prefer to use the word sequence) of even numbers are consecutive even numbers, or even more generally an arithmetic sequence. If we are not given any information about the sequence other than that each member happens to be even, there is no formula for that other than the fact that you can factor out the 2 from each member and add up the halves, then multiply by 2: 2a + 2b + 2c = 2(a + b + c). If the even numbers are an arithmetic sequence, you can use the formula for the sum of an arithmetic sequence. Similarly if they are a geometric sequence.

12, 6, 0, -6, ...

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