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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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, ...