Please help....

Question

A new Wholesale Club opened in town and gaining in popularity. After its great opening, every day the number of shoppers is 5% more than the number of shoppers the day before. If the total number of shoppers over the first 10 days is 1258, how many shoppers were at the Wholesale club on the opening day?

Answer 1

Let's say for example that there were 50 shoppers the first day.

Then on day 2 there would be 1.05(50). (a 5% increase from the day before).

Then on day 3 there would be 1.05(1.05)(50) or 1.05²(50)

Let's do one more day:

On day 4: 1.05³(50.

The point is this is a geometric sequence where r = 1.05 and a₁=50.

The total number of shoppers is the sum of the number on each day (a geometric series).

There is a formula for the sum of a finite geometric series:

S_n= a₁[(1-rⁿ)/(1-r)]

In this particular case S_n=1258 and r = 1.05.

So,

1258=a₁[(1 - 1.05¹⁰)/(1 - 1.05)]

a₁=1258(1 - 1.05)/(1 - 1.05¹⁰)

desmos.com/calculator/rtnggahpgf

Not an ideal problem since the number of shoppers on each day is not a whole number.

Answer 2

This turns out to be a geometric series.

g₂=1.05g₁

g₃=1.05g₂

g₄=1.05g₃

:

g₁₀=1.05g₉

Total = S_n = g₁∑ⁿ_i=1 r^i-1 = g₁(rⁿ-1)/(r-1)

r = 1.05

S₁₀ = 1258 = g₁∑¹⁰_i=11.05^i-1= g₁(r¹⁰-1)/(r-1) = g₁(1.05¹⁰-1)/0.05 ≈ g₁(12.58)

g₁ = 1258/12.58 = 100 shoppers on the first day

Answer 3

Let's designate our unknown, number of shoppers on the opening day, with the letter a

So, the first day we have a shoppers,

the second day we have a+a(5/100) = a(1+5/100) = a(1.05)

the third day we have

[a+a(5/100)] + [a+a(5/100)] [5/100] =a+a(5/100)+a(5/100)+a(5/100)²

which is equal to a (1+5/100)² =a(1.05)²

and so forth…

The total accumulated number of shoppers would be a+ a(1.05)+ a(1.05)²+…

This expression can be recognized as a geometric series whose ratio is r=1.05, being a the first term in the sequence, and n is the number of terms. In our case n=10

The formula for a finite geometric series Sn is given by

S_n=(a(1-rⁿ)) / (1-r)

We need to solve for a. We were told that the total number of shoppers over the first 10 days is 1258, hence S_n=1258

Thus,

a=(S_n (1-r)) / (1-rⁿ )

a=1258(1-1.05) / (1-(1.05)¹⁰ ) =100.01, rounding down to a=100

Therefore, the number of shoppers that visited the Wholesale Club during the first day was 100.

Hope that helps