SOLUTION: The numbers 2, b, c, d, 72 are listed in increasing order so that 2, b, c form an arithmetic sequence, b, c, d form a geometric sequence, and c, d, 72 form a harmonic sequence. Wha

Algebra ->  Sequences-and-series -> SOLUTION: The numbers 2, b, c, d, 72 are listed in increasing order so that 2, b, c form an arithmetic sequence, b, c, d form a geometric sequence, and c, d, 72 form a harmonic sequence. Wha      Log On


   

Question 1199566: The numbers 2, b, c, d, 72 are listed in increasing order so that 2, b, c form an arithmetic sequence, b, c, d form a geometric sequence, and c, d, 72 form a harmonic sequence. What is the value of b + c?
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Let x be the common difference in the arithmetic sequence. Then....

2, b, and c form an arithmetic sequence, so
b+=+2%2Bx
c+=+2%2Bb+=+2%2B2x

b, c, and d form a geometric sequence, so
bd+=+c%5E2
d+=+c%5E2%2Fb

c, d, and e form a harmonic sequence, so
2%2Fd=1%2Fc%2B1%2Fe
2%2Fd-1%2Fc=1%2Fe
%282c-d%29%2Fcd=1%2Fe
e=cd%2F%282c-d%29

So we have
b=x%2B2
c=2x%2B2
d=c%5E2%2Fb=%282x%2B2%29%5E2%2F%28x%2B2%29


We need to have e = 72. Simplify the expression for e, starting by multiplying numerator and denominator of the fraction by (x+2).

e=%28%282x%2B2%29%5E3%29%2F%28%282%282x%2B2%29%28x%2B2%29-%282x%2B2%29%5E2%29%29

e=%28%282x%2B2%29%5E3%29%2F%28%282x%2B2%29%282%28x%2B2%29-%282x%2B2%29%29%29

e=%282x%2B2%29%5E3%2F%28%282x%2B2%29%282%29%29

e=%282x%2B2%29%5E2%2F2+=+4%28x%2B1%29%5E2%2F2+=+2%28x%2B1%29%5E2

Then solve for e=72.

2%28x%2B1%29%5E2=72
%28x%2B1%29%5E2=36
x%2B1=6
x=5

And we have for the sequence of numbers:
2
b = 2+5 = 7
c = 7+5 = 12
d = 12^2/7 = 144/7
e = (12(144/7))/(24-144/7) = (1728/7)/(24/7) = 72

ANSWER: b+c = 7+12 = 19