NIMCET 2008 — Mathematics PYQ

Since $a,b,c$ are in Arithmetic Progression (AP), the middle term is the arithmetic mean of the other two:

Or, more simply, the difference between consecutive terms is constant ($d$):

• $b=a+d$

• $c=a+2d$

2. Examine the Given Terms

We need to determine the nature of the sequence:

3. Test for Geometric Progression (GP)

A sequence is in GP if the square of the middle term equals the product of the first and last terms ($T_2^2=T_1\cdot T_3$). Let's check this:

Calculate $T_1\cdot T_3$:

Using exponent rules ($e^x\cdot e^y=e^{x+y}$):

Finding a common denominator for the exponent:

Calculate $T_2^2$:

Using exponent rules ($(e^x{)}^y=e^{xy}$):

4. Compare the Results

We know from the properties of AP that $2b=a+c$. Substituting this into our $T_2^2$ equation:

Comparing this to our result for $T_1\cdot T_3$:

Since $T_2^2=T_1\cdot T_3$, the terms satisfy the condition for a Geometric Progression.

Conclusion

The sequence formed by the exponential terms is a Geometric Progression because the square of the middle term is equal to the product of the first and third terms.

Correct Option: (b)