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

2 Answers
Feb 8, 2017

Answer:

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#

Feb 8, 2017

Answer:

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.!