Question

These are questions i notice a pattern with, but don't know how to approach. I need help!.

An equestrian wants buy a horse for $10,000. He made a deal with the salesman to pay for the nails in the horseshoes. He paid 1 cent for the first nail, 2 cents for the second nail, 4 cents for the third and so on. Each horseshoe is fastened by 5 nails. Did he make a good deal?

A workman agreed to work under the following conditions: His salary for the first day of work will be $1, for the second day of work $2, and for the third day of work $4, and so on. How long does he have to work to earn 4095?

Answer 1

The equestrian overbid the horse by $485.75.

It takes the workman 12 days.

Explanation:

Each nail costs 2 times more than the previous nail. So this is a geometric sequence where the first term is 1 and the common ratio is 2.

a₁ = 1

r = 2

There are 5 nails per horseshoe, and 4 horseshoes on a horse, for a total of 20 nails. The sum of the first n terms of a geometric series is:

S = a₁ (1 − rⁿ) / (1 − r)

Plugging in values:

S = 1 (1 − 2²⁰) / (1 − 2)

S = 1,048,575

The nails cost 1,048,575 cents, or $10,485.75. This is more than the cost of the horse, so the equestrian did not make a good deal.

Like before, this is a geometric sequence where the first term is 1 and the common ratio is 2.

The sum of the first n terms is:

S = a₁ (1 − rⁿ) / (1 − r)

Plugging in values:

4095 = 1 (1 − 2ⁿ) / (1 − 2)

4095 = 2ⁿ − 1

4096 = 2ⁿ

n = 12

It takes the worker 12 days.