An entrepreneur starts a company with an operating budget of 100,000 in its first year. Fo… — ACT

An entrepreneur starts a company with an operating budget of $100,000$ in its first year. For each of the next 4 years (years 2 through 5), the budget increases by a constant amount of $d$ dollars each year. For years 6 through 8, the budget increases geometrically, where the budget in year 6 is $1.5$ times the budget in year 5, and the budget increases by $50\%$ each year thereafter. If the total operating budget over the first 8 years is $1,597,500$ dollars, what is the value of $d$ ?

A
$4,300$
$10,000$ Answer
C
$17,500$
D
$33,900$
E
$38,500$

Answer

The common difference is

10,000

The correct answer is found by setting up the sum of the first 5 years of the arithmetic sequence and the subsequent 3 years of the geometric sequence. Summing the expressions for all 8 years gives

1,212,500 + 38.5d

. Equating this expression to the total budget of

1,597,500

and solving for

d

yields

10,000

Step-by-Step Solution

Express the budget for the first 5 years as an arithmetic sequence and find their sum.

S_5 = 100,000 + (100,000 + d) + (100,000 + 2d) + (100,000 + 3d) + (100,000 + 4d) = 500,000 + 10d

The budget starts at

100,000

in year 1 and increases by a constant amount

d

each year through year 5.

Express the budgets for years 6 through 8 as a geometric sequence starting from

1.5

times the year 5 budget.

Year 6:

1.5(100,000 + 4d) = 150,000 + 6d

; Year 7:

1.5(150,000 + 6d) = 225,000 + 9d

; Year 8:

1.5(225,000 + 9d) = 337,500 + 13.5d

. The sum of these 3 years is

712,500 + 28.5d

The budget increases by a factor of

1.5

(or

50\%

) each year starting from year 6.

Combine the sums of both sequences to represent the total 8-year budget and solve for

d

Total =

(500,000 + 10d) + (712,500 + 28.5d) = 1,212,500 + 38.5d

. Setting this equal to the given total:

1,212,500 + 38.5d = 1,597,500 \implies 38.5d = 385,000 \implies d = 10,000

The total budget over the 8 years is the sum of the budgets of the individual years.

Key Concept

Combining arithmetic and geometric sequences in multi-step word problems.

Alternative Method

Instead of calculating each geometric year sequentially, the sum of the geometric sequence for years 6 to 8 can be calculated using the geometric series sum formula

S_n = a_1 \frac{1 - r^n}{1 - r}

with

a_1 = 1.5(100,000 + 4d)

and

r = 1.5

over

n = 3

terms.

Estimated Time:2m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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