NIMCET 2021 — Mathematics PYQ

1. Identify the Terms of the G.P.

When we insert four geometric means ($G_1,G_2,G_3,G_4$) between $2$ and $64$, we form a Geometric Progression:

In this sequence:

• First term ($a$) = $2$

• Last term ($l$ or $a_n$) = $64$

• Total number of terms ($n$) = $2\text{ (original)}+4\text{ (means)}=6$

2. Find the Common Ratio ($r$)

The formula for the $n^{th}$ term of a G.P. is:

Substitute the known values:

Divide both sides by $2$:

Since $32=2^5$, we have:

3. Calculate the Geometric Means

Now, we find the four means by multiplying the first term by the common ratio successively:

• $G_1=a\cdot r=2\cdot 2=4$

• $G_2=a\cdot r^2=2\cdot 2^2=8$

• $G_3=a\cdot r^3=2\cdot 2^3=16$

• $G_4=a\cdot r^4=2\cdot 2^4=32$

Conclusion

The four geometric means are $4,8,16,\text{ and }32$.

Correct Option: B) $4,8,16,32$