Final answer:
The value of 5³ * 5⁴ * 5⁵ * ... * 5ᵓ, where k is an integer greater than 1, can be found by adding the exponents and simplifying. The resulting value is 5 to the power of the sum of an arithmetic series with terms from 3 to k.
Step-by-step explanation:
To find the value of 5³ * 5⁴ * 5⁵ * ... * 5ᵓ, where k is any integer greater than 1, we use the properties of exponents. When multiplying powers with the same base, we simply add the exponents.
Therefore, the expression simplifies to 5⁵+⁴+⁵+...+ᵓ. Summing the exponents from 3 up to k is the same as finding the sum of an arithmetic series. The sum of this series can be found using the formula S = (n/2)(first term + last term), where n is the number of terms. In this case, n is k-2, the first term is 3, and the last term is k.
Therefore, the sum of the exponents is (k-2+1)/2 * (3+k). Simplifying this we get (k+1)/2 * (3+k). So the final value of the expression is 5⁵×(k+1)/2×(3+k).