Question

Find the number of terms in the sequence 1,-4,16,...,65536

Answer 1

9 terms

Explanation:

The sequence given is a geometric sequence

In a geometric sequence, the nth term of the sequence can be found by the formula

`a_n = a_1r^(n-1)`

`\text{where }\\\\ a_n = \text {nth term}\\\\\text{$a_1 = $ first term}\\\\\text{$r = $ common ratio}`

In the given sequence,

a₁ = 1

r = -4

aₙ = 65536

So we get the relation:
65536 = 1· (-4)ⁿ⁻¹

65536 = (-4)ⁿ⁻¹

It is clear that n-1 has to be even so that the power of 4 is positive.
Substituting x = n -1 where x is even gives us
4ˣ = 65536

If we take logarithms to the base 4 on both sides we get

=> x =
`\log_465536 = 8\\\\`

Since x = n - 1 and x = 8, n = 9

So the 9th term in the series is 65536