Question

Find an explicit formula for each sequence given below. (Assume for each that the domain is Z⁺ = {1,2,3.....}

a. 1,-1/2,3/4,-5/6,7/8,......
b. 4,-1/8,16,-1/32,64,-1/128,....

Answer 1

Sequence (a) has the nth term given by the formula
`a_n = (-1)^(n+1)*(2n - 1)/(2n)`y the formula
`b_n = (-1/2)^(n-1)*(2^(2n-2))`s that can be described mathematically.

Step-by-step explanation:

The question involves finding explicit formulas for two given sequences. For sequence (a) 1, -1/2, 3/4, -5/6, 7/8,..., this sequence can be described by the nth term formula
`a_n = (-1)n+1*(2n - 1)/(2n)`nates between positive and negative values and increases by 2 with each term, while the denominator increases by 2 and is always even.

For sequence (b) 4, -1/8, 16, -1/32, 64, -1/128,..., we can use the formula
`b_n = (-1/2)n-1*(22n-2)`la captures the pattern of alternating between positive and negative terms, with positive terms being powers of 2 and negative terms being negative reciprocals of powers of 2.