Final answer:
The number of terms in the geometric progression sequence with a first term of 5, a last term of 1280, and a sum of 2555 is 9. The common ratio of the GP is determined as 2 by dividing the last term by the first term and identifying its root, and the formula for sum of GP is applied to find the total number of terms.
Step-by-step explanation:
To find the number of terms in a geometric progression (GP) where the first term is 5 and the last term is 1280, and the sum of all terms is 2555, we can use the formula for the sum of a finite GP: S = a*(1-r^n) / (1-r), where S is the sum of the GP, a is the first term, r is the common ratio, and n is the number of terms.
Given: a = 5, and S = 2555.
We need to find the common ratio r by using the last term of the GP. The last term is given by T_n = a*r^(n-1). In this case, T_n = 1280.
Solving for r, we get 1280 = 5*r^(n-1). Dividing both sides by 5 gives 256 = r^(n-1). Since 256 is a power of 2 (2^8), the common ratio r is 2 and n-1 = 8, so n = 9.
We can now confirm that this number of terms gives the correct sum by substituting the values into the sum formula: 2555 = 5*(1-2^n) / (1-2), which leads to 2555 = 5*(1-2^9) / (1-2), confirming that 9 is the correct number of terms.