How to know if a raise is arithmetic or geometric?what is equation for each (to find amount earned on Xth year)?How to find total amount earned during X years?
$X = arithmeticX% = geometricarithmetic:an = $salary + (n-1)$raisegeometric:an = $salary( 1 + raise%as decimal) ^n-1arithmetic: X years^E (simplified equation from above) i=1S = n(a1+an/2)geometric:X years^ E (simplified equation from above) i=1S = a(1-r^n/1-r)
To determine if a raise is arithmetic or geometric, you need to look at the pattern of increase in the salary.
Arithmetic growth refers to a steady, constant increase in the salary. For example, if an employee earns $1000 in the first year, and then receives a raise of $100 each year, their salary would be $1100 in the second year, $1200 in the third year, and so on. We can determine this by using the equation:
Salary = Starting salary + (Number of years – 1) * Rate of increase
Geometric growth, on the other hand, refers to a pattern of increasing growth in the salary. For example, if an employee earns $1000 in the first year, and then receives a raise of 10% each year, their salary would be $1100 in the second year, $1210 in the third year and so on. We can determine this by using the equation:
Salary = Starting salary * (1 + Rate of increase)^Number of years
To find the amount earned on the Xth year, simply plug the value of X into the appropriate equation and solve for Salary:
For arithmetic growth:
Salary = Starting salary + (X – 1) * Rate of increase
For geometric growth:
Salary = Starting salary * (1 + Rate of increase)^X
To find the total amount earned during X years, use the appropriate Summation formula:
For arithmetic growth:
Total earnings = (Number of years/2) * [2*Starting salary + Rate of increase*(Number of years-1)]
For geometric growth:
Total earnings = Starting salary * [(1 + Rate of increase)^Number of years – 1] / Rate of increase
Keep in mind that in practice, it is possible for a raise to be neither strictly arithmetic nor geometric, and therefore require a more complex formula for calculation.
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