Statement 1 `4,8,16,` are in GP and 12,16,24 are in HP.
Statement 2 If middle term is added in three consecutive terms of a GP, resultant will be in HP.

Options

A. Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1
B. Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1
C. Statement 1 is true, Statement 2 is false
D. Statement 1 is false, Statement 2 is true

Answer

A

Explanation

If a,b,c are in GP.
Then, `b^(2)=ac`
If middle term is added, then `a+b,2b " and " c+b` are in GP.
`(I-II)/(II-III)=(a+b-2b)/(2b-(c+b)) " " [" here ",I=a+b,II=2b,III=c+b]`
`=(a-b)/(b-c)=(ab-b^(2))/(b^(2)-bc)=(ab-ac)/(ac-bc)" " [:.b^(2)=ac]`
`=(a(b-c)(a+b)(b+c))/(c(a-b)(a+b)(b+c))`
`=(a(b^(2)-c^(2))(a+b))/(c(a^(2)-b^(2))(b+c))`
`=(a(ac-c^(2))(a+b))/(c(a^(2)-ac)(b+c)),(a+b)/(b+c)=(I)/(II)`
Hence, `a+b,2b,b+c` are in HP.
Hence, both statements are true and Statement 2 is correct explanation for Statement 1.