Answer :
Only the nth term of the AP is necessarily equal to the nth term of the GP, hence correct answer is 4) The nth term of the AP is equal to the nth term of the GP.
mth, nth, and pth terms are equal:
We are given that the mth, nth, and pth terms of both the AP and GP are equal and are x, y, and z respectively. This essentially means:
mth term of AP = x = mth term of GP
nth term of AP = y = nth term of GP
pth term of AP = z = pth term of GP
Why other options are not necessarily true:
Common difference vs. common ratio: The fact that some terms are equal doesn't guarantee the common difference of the AP (which is added between terms) is equal to the common ratio of the GP (which is multiplied between terms).
Sum of terms: The sum of an AP and GP depends on the specific values of first terms, number of terms, common difference/ratio, and the terms being summed. Just knowing a few equal terms is not enough to determine if the sums are equal.
Product of terms: Similar to the sum, the product of terms in an AP and GP depends on the specific terms involved. Just knowing a few equal terms is not enough to determine if the products are equal.
Why nth terms are necessarily equal:
The fact that the nth terms of both the AP and GP are equal (y = nth term of AP = nth term of GP) implies that at the nth position, the value is the same in both progressions. This holds true regardless of the specific values of the first terms, common difference/ratio, or other terms in the sequences.
Therefore, only option 4 guarantees that the nth term of the AP is equal to the nth term of the GP based on the given information.
