Respuesta :
If the employee had a final salary of 9000 pa after 10 years and if the total sum of all her salaries was 65000, then her initial salary was 2818.18
Given: An employee received fixed annual increments. His final salary after 10 years is 9000. The sum of all of his salaries in the 10 years is 65000. To find his initial salary.
Let's solve the problem:
We assume that the employee's initial salary was A.
Let aₙ denote the employee's salary where n is the year number. [ n = 0, 1, 2, .... 10 ]
So, a₀ = A (initial salary)
Also, the employee received a fixed annual increment each year.
As it is only mentioned that the annual increment was fixed, so let us assume that the fixed increments per year are d
So after n number of years, the employee's salary can be calculated as:
aₙ = a₀ + nd [ n = 0, 1, 2, .... n ] where
aₙ = employee's salary after n years
a₀ = employee's initial salary (also A)
n = number of years
d = fixed increment per year
Hence we can write aₙ as
aₙ = A + nd [a₀ = A]
It is also given that the final salary after 10 years is 9000
That is, a₁₀ = 9000
Sum of all the salaries of the employee after 10 years is 65000.
Let us denote the sum of all salaries after 10 years as S₁₀
Therefore S₁₀ = 65000
In Arithmetic progression, the formula for summation of all terms of the AP when the initial and the last values are given is:
Sₙ = n / 2 (a₀ + aₙ)
Where Sₙ is the summation of n terms
n = Total number of terms
a₀ = initial value
aⁿ = final or last value
Here n = 11 because n started from the 0th year to the 10th year. So 11 years.
Therefore, S₁₀ = n / 2 ( a₀ + a₁₀ )
So putting the values we get:
65000 = 11 / 2 ( A + 9000)
or
11 / 2 ( A + 9000) = 65000
Multiplying both sides by 2 / 11, we get
11 / 2 ( A + 9000) × 2 / 11 = 65000 × 2 / 11
= A + 9000 = 11818.18
Subtracting 9000 from both sides :
A + 9000 - 9000 = 11818.18 - 9000
= A = 2818.18
Therefore the initial value is 2818.18
Hence if the employee had a final salary of 9000 pa after 10 years and if the total sum of all her salaries was 65000, then her initial salary was 2818.18
Know more about "arithmetic progression" here: https://brainly.com/question/14278705
#SPJ9
