Question

How do you find the missing terms of the geometric sequence:2, __, __, __, 512, ...?

Answer 1

There are four possibilities:

`8, 32, 128`

`-8, 32, -128`

`8i, -32, -128i`

`-8i, -32, 128i`

Explanation:

We are given:

`{ (a_1 = 2), (a_5 = 512) :}`

The general term of a geometric sequence is given by the formula:

`a_n = a*r^(n-1)`

where `a` is the initial term and `r` the common ratio.

So we find:

`r^4 = (ar^4)/(ar^0) = a_5/a_1 = 512/2 = 256 = 4^4`

The possible values for `r` are the fourth roots of `4^4`, namely:

`+-4`, `+-4i`

For each of these possible common ratios, we can fill in `a_2, a_3, a_4` as one of the following:

`8, 32, 128`

`-8, 32, -128`

`8i, -32, -128i`

`-8i, -32, 128i`

Answer 2

The Missing Terms are, `8, 32, and, 128.`

Explanation:

Let `r` be the Common Ratio of the given Geo. Seq. denoted, by

`{a_n}_(n in NN).`

Then, `a_1=2, and, a_5=512.`

But, we know that, `a_n=a_1*r^(n-1), n in NN.`

`:. a_5=512:.a_1*r^(5-1)=512:.2*r^4=512:.r^4=256=4^4`

`:. r=4`.

Hence, the reqd. missing terms, known as, Intermediate

Geometric Means, are,

`a_2=2*4^(2-1)=2*4=8, a_3=2*4^2=32, &, a_4=128`.

Enjoy Maths.!