Q54. If the Geometric mean of the 7th and 10th terms is (3) ^ (1/4) If the GP consists of 16 terms, Find the Product of all terms. Ops: A 243 B. 81 C01 D. 027

The question is asking for the product of all terms in a geometric progression (GP) given that the geometric mean of the 7th and 10th terms is (3)^(1/4) and the GP consists of 16 terms.

Step 1: Understand the properties of a geometric progression In a geometric progression, the ratio of any two consecutive terms is constant. This ratio is also equal to the nth root of the ratio of the nth term to the first term.

Step 2: Use the given information We are given that the geometric mean of the 7th and 10th terms is (3)^(1/4). The geometric mean of two numbers a and b is the square root of their product, i.e., sqrt(ab). Therefore, we can write the equation as follows:

sqrt(a7 * a10) = (3)^(1/4)

where a7 and a10 are the 7th and 10th terms of the GP respectively.

Step 3: Simplify the equation Since a7 and a10 are terms in a GP, we can write them in terms of the first term a and the common ratio r as follows:

a7 = ar^6 and a10 = ar^9

Substitute these into the equation:

sqrt(ar^6 * ar^9) = (3)^(1/4)

Simplify to get:

ar^(15/2) = (3)^(1/4)

Step 4: Find the product of all terms The product of all terms in a GP is given by the nth root of the product of the nth power of the first term and the nth power of the last term. Since we have 16 terms, the product is:

P = (a^16 * r^120)^(1/16)

Substitute ar^(15/2) = (3)^(1/4) into the equation:

P = ((3)^(1/4))^16 * r^120)^(1/16)

Simplify to get:

P = 3 * r^120

Since we don't have the value of r, we can't find the exact value of P. Therefore, the question seems to be incomplete or incorrect.