Question

If f(x) = 2^x + 1, then f(-3), f(1) and f(3) - 1 are in 
1. H.P.
2. G.P.
3. A.P.
4. None of the above

Answer 1

Correct Answer - Option 2 : G.P.

Concept:

• If a, b, c are in A.P ⇔ \( \rm b =\frac{a+c}{2} \)   
• If a, b, c are in G.P ⇔  b² = ac
• If a, b, c are in H.P ⇔\( \rm b =\frac{2ac}{a+c} \)  

 

Calculation:

Given: f(x) = 2x + 1

f(-3) = 2-3 + 1 = \(\frac{1}{2^3}+1=\frac18+1=\frac98\)

f(1) = 2 + 1 = 3

f(3) - 1= 2³ + 1 - 1 = 8

Now, we have \(\frac98\), 3 and 8

Clearly, (3)² = 8 × \(\frac98\)

So, [f(1)]² = f(-3) × (f(3) - 1)

We know, if a, b, c are in G.P then b2 = ac

∴ f(-3), f(1) and (f(3) - 1) are in G.P.

Hence, option (2) is correct.