A guy saves $100 in his first year of work and each year saves $20 more than in the preceeding year. In how many years will he save $5800?

Answer 1

Let's say the number of years he saves the money is x.
The amount saved in the first year is $100.
From the second year, the amount saved increases by $20 each year.
So the amount saved in the second year is $100 + $20 = $<<100+20=120>>120.
The amount saved in the third year is $120 + $20 = $<<120+20=140>>140.
And so on until the xth year, which is $100 + ($20 * (x-1)) = $100 + $20x - $20 = $80 + $20x.
The total amount of money saved in x years is $100 + $120 + $140 + ... + ($80 + $20x).
This sum can be represented as the sum of an arithmetic series with first term $100, last term $80 + $20x, and common difference $20.
The formula for the sum of an arithmetic series is S = (n/2)(a + l), where S is the sum, n is the number of terms, a is the first term, and l is the last term.
In this case, the sum is $5800, the first term is $100, and the last term is $80 + $20x.
So we have the equation (x/2)($100 + ($80 + $20x)) = $5800.
Simplifying this equation gives (x/2)(2*$100 + ($80 + $20x)) = $5800.
Multiplying through the brackets gives (x/2)(2*$100 + $80 + $20x) = $5800.
Simplifying the inside of the brackets gives (x/2)(2*$100 + $80 + $20x) = $5800.
Multiplying through the brackets gives (x/2)*($200 + $80 + $20x) = $5800.
Simplifying the inside of the brackets gives (x/2)*($280 + $20x) = $5800.
Multiplying through the brackets gives (x/2)*$280 + (x/2)*$20x = $5800.
Simplifying the left side gives (x*$280)/2 + ($20x^2)/2 = $5800.
Simplifying the left side gives ($140x + 10x^2)/2 = $5800.
Multiplying through by 2 gives $140x + 10x^2 = $11600.
Rearranging the equation gives 10x^2 + $140x - $11600 = 0.
Dividing through by 10 gives x^2 + $14x - $1160 = 0.
Using the quadratic formula gives x = (-b +/- sqrt(b^2 - 4ac))/(2a).
In this case, a = 1, b = $14, and c = -$1160.
Substituting these values into the quadratic formula gives x = (-$14 +/- sqrt(($14)^2 - 4(1)(-$1160)))/(2(1)).
Simplifying the inside of the square root gives x = (-$14 +/- sqrt($196 + 4640))/(2(1)).
Simplifying the inside of the square root gives x = (-$14 +/- sqrt($4836))/(2(1)).
Simplifying further gives x = (-$14 +/- $69.6)/(2(1)).
Splitting into two separate solutions gives x = (-$14 + $69.6)/(2(1)) or x = (-$14 - $69.6)/(2(1)).
Simplifying further gives x = $55.6/2 or x = -$83.6/2.
So we have two possible solutions: x = $27.8 or x = -$41.8.
Since time cannot be negative, we can ignore the x = -$41.8 solution.
Therefore, the only valid solution is x = $27.8.
Therefore, it will take him 27.8 years to save $5800.
However, since it is not possible to save part of a year, we round up to the next whole number.
Therefore, it will take him 28 years to save $5800. Answer: \boxed{28}.