Expert: Josh - 5/25/2006

Question
Replace the ? with appropriate number-
Q.1) 18, 14, 12, 11, ?, 10.25
Q.2) 5, 23, 113, 523, ?
Q.3) 8, 6, 5, 4.5, ?
Q.4) 5. 24, 63, 128, 225, ?

I solved the first and the third questions but couldn't do the rest, Please help me in these.

Answer
Q2 involves a logarithmic function.
It is hardly a fair question to ask, since it is not obvious unless you plot a graph. Then, it will reveal a linear relationship in the logarithmic domain.

Let (x[1],x[2],x[3],x[4])=(5,23,113,523).
Taking log_2(x[i]), we get (2.32192809488736,4.52356195605701,6.82017896241519,9.03066713624694)

It has a slope of m=(log_2(x[2])-log_2(x[1]))=2.20163386116965 with respect to i.

The next data point log_2(x[5])=log_2(x[4])+m*1=
11.2323009974166. Taking the anti-log, i.e., raising 2 to the power of 11.2323009974166, we get x[5]=2405.8.

Q4 If you plot sqrt(x(i)) versus i, you will get a roughly (not exactly) linear relationship.

Like before, the slope is approximately m=sqrt(x(2))-sqrt(x(1)). Using linear prediction, sqrt(x(6))~=sqrt(x(5))+m. Thus, x(6)~=[sqrt(x(5))+m]^2 = 311.98

Rounding this to the nearest integer, a reasonable value for x(6) is 312.