Consider the sequence −2/3, 4/5, −8/7, 16/9, . . . , where a1 = −2/3.

Question

(a) Write down the next two terms in the sequence, and find an exact formula for it

(b) Select all the properties that are true for this sequence, and support your answer with an explanation: arithmetic, geometric, bounded, monotonic.

Answer 1

-32/11, 64/13 are the next two terms

a1 = -2/3,

a2 = 4/5

a3 = -8/7

a4 = 16/9

a5 = -32/11

a6 = 64/13

a7 = -128/15

an = ((-2)^n)/(2n+1)

It's an unbounded sequence with no limit.

It oscillates in sign, diverging.

It's neither arithmetic or geometric.

but

the numerator is geometric with a common ratio, r = -2

the denominator is arithmetic with a common difference, d = 2

Answer 2

The numerator appears to be (-2)ⁿ and the denominator appears to be 2n+1.

The absolute value of the numerator is larger than the numerator for large n, i.e. the sequence is unbounded and alternating (in sign) for large n.