A Man earn a salary of 2500 per annum. If a rise of 500 is given at the end of the year. Find the total amount the man will earn in 8 years (AP)
If a man earns a salary of 2500 per annum and given a raise of 500 at the end of the year, it means that his salary increases by 500 every year. This is an example of an arithmetic progression (AP) where the first term (a) is 2500 and the common difference (d) is 500.
To find the total amount the man will earn in 8 years, we need to find the sum of the first 8 terms of this arithmetic progression.
The formula for the sum of the first n terms of an arithmetic progression is given by:
S = n/2 [2a + (n-1)d]
where:
S = sum of the n terms
n = number of terms
a = first term
d = common difference
Here, n = 8, a = 2500 and d = 500. Plugging these values into the formula, we get:
S = 8/2 [2(2500) + 7(500)]
S = 4 [5000 + 3500]
S = 4 * 8500
S = 34000
Therefore, the man will earn a total of 34000 in 8 years.
To find the total amount the man will earn in 8 years, we need to calculate the arithmetic progression (AP) for those 8 years.
First, let's find the common difference (d) in the arithmetic progression. The rise of 500 given at the end of the year is added to the initial salary of 2500. So, the total salary for the first year is 2500 + 500 = 3000.
Now, we can calculate the common difference:
d = 3000 - 2500 = 500
Since the increment of 500 is given at the end of each year, the total salary for the second year will be 3000 + 500 = 3500.
Using the formula for the nth term of an arithmetic progression:
aₙ = a + (n-1)d
where:
aₙ is the nth term,
a is the first term,
n is the number of terms,
and d is the common difference,
we can find the salary for each year by substituting the respective values.
So, the salary for the 8 years will be:
a₁ = 3000
a₂ = a₁ + d = 3500
a₃ = a₂ + d = 4000
a₄ = a₃ + d = 4500
a₅ = a₄ + d = 5000
a₆ = a₅ + d = 5500
a₇ = a₆ + d = 6000
a₈ = a₇ + d = 6500
To find the total amount earned in 8 years, we add up all the salaries:
Total = a₁ + a₂ + a₃ + a₄ + a₅ + a₆ + a₇ + a₈
Total = 3000 + 3500 + 4000 + 4500 + 5000 + 5500 + 6000 + 6500
Total = 36000
Therefore, the man will earn a total of 36000 in 8 years.
To find the total amount the man will earn in 8 years, we need to find the annual salary for each year and then sum them up.
Given:
Initial salary (first year) = $2500
Rise at the end of the year = $500
Since the salary follows an arithmetic progression (AP), we can use the formula for the nth term of an AP to find the salary for each year:
An = A1 + (n - 1)d
Where:
- An is the nth term (salary for a particular year)
- A1 is the initial term (initial salary)
- n is the year number
- d is the common difference (the rise at the end of each year)
Let's calculate the salary for each year using the given information:
Year 1: A1 = $2500 (Initial salary)
Year 2: A2 = A1 + d = $2500 + $500 = $3000
Year 3: A3 = A2 + d = $3000 + $500 = $3500
and so on...
To find the total amount earned over 8 years, we need to sum up the salaries for each year. Since this is an AP, we can find the sum using the formula for the sum of a finite arithmetic series:
Sn = (n/2)(A1 + An)
Where:
- Sn is the sum of the salaries for n years
- A1 is the initial term (initial salary)
- An is the nth term (salary for the last year)
Using the values we have calculated, let's find the total amount earned in 8 years:
n = 8 (8 years)
Sn = (8/2)(A1 + An)
= 4(A1 + An)
= 4($2500 + An)
To find An, the salary for the 8th year, we use the formula for the nth term of an AP:
An = A1 + (n - 1)d
= $2500 + (8 - 1)($500)
= $2500 + 7($500)
= $2500 + $3500
= $6000
Substituting the values into the formula for the sum:
Sn = 4($2500 + $6000)
= $4(8500)
= $34000
Therefore, the man will earn a total of $34,000 in 8 years.