Every week the sales of a band’s album drop by the same percentage relative to the previous week’s sales.

Question

In its first week an album sells 8,000 copies. In its seventh week an album sells 2,800 copies. Assuming this trend continues, calculate the total number of albums which will be sold. Round your answer to the nearest album.

Answer 1

 

This can be worked using geometric sequences.

 

The base formula is  ... a_n = a₁ * r^{n - 1}

 

   where "a₁" is the first term (8000) and "a_n"is the nth term, and "r" is the common ratio

 

We can substitute a₁ =8000, a₇ = 2800, and n=7, to solve for "r"

 

   2800 = 8000(r)^7-1

   2800 = 8000(r)⁶

   2800/8000 = r⁶         Divide by 8000 on both sides

   .83948191 = r           Take the 6th root of both sides

 

So our general formula for the nth term of the sequence is:

 

    a_n = 8000(.83948191)^n-1

 

Since we were instructed to round to the nearest album, I set a_n to .5 ... so when the term whittles down to .5 of an album, we want sales to stop at that point.  When we substitute this in, we can solve for "n" to obtain the last term in the sequence.  we could also set the value to 1 and get a slightly different answer.  I was not sure which value to use.  I chose .5.

 

    .5 = 8000(.83948191)^n-1

    .5/8000 = .83948191^n-1                                                         Divide by 8000

    .0000625 = .83948191^n-1

 

    ln (.0000625) = ln (.83948191)^n-1                              Take natural log of both sides

    ln (.0000625) = (n - 1) * ln (.83948191)                    Log powers rule

    ln (.0000625) = n * ln (.83948191) - ln (.83948191)   Distribute

 

    ln (.0000625) + ln (.83948191) = n * ln (.83948191)  Solve for n

    [ln (.0000625) + ln (.83948191)]/ln (.83948191) = n

 

    n = 56.32

 

    Thus it will take 56 weeks for the albums sales to fall to 1 album.

 

To find the number of albums sold, we use the formula for the sum of the first n terms of a geometric sequence

 

    s_n = a₁(1 - r)ⁿ/(1 - r)

 

     s_n  = 8000(1 - .83948191)⁵⁶/(1 - .83948191)

     s_n  = 49,836

 

Thus, 49,836 albums are sold in 56 weeks