construct geom. seq. w/ 13 pts, arith. seq. w/ 16 pts

[leyva2389]

Suppose we are given an interval (a,b) and are asked to construct a geometric sequence of 13 points where a₀=a and a₁₃=ar¹²=b and an arithmetic sequence of 16 points where a₀=a and a₁₅=a+15d=b. If we construct these sequences and find that ar⁶=a+7d, what is the length of the interval?

Just plain stuck! Where do I even start? Please help.

 

[DrPhil]

Suppose we are given an interval (a,b) and are asked to construct a geometric sequence of 13 points where a₀=a and a₁₃=ar¹²=b and an arithmetic sequence of 16 points where a₀=a and a₁₅=a+15d=b. If we construct these sequences and find that ar⁶=a+7d, what is the length of the interval?

Just plain stuck! Where do I even start? Please help.

For the arithmetic sequence, the step distance between points is d. Solve the equation \(\displaystyle a+15d = b\) for \(\displaystyle d\).

Likewise for the geometric sequence, solve for the ratio \(\displaystyle r\) as a function of the endpoints.

Show us what you get.

Then plug those expressions for r and d into the equation ar⁶=a+7d. Your task is to (b-a).

 

[pka]

Suppose we are given an interval (a,b) and are asked to construct a geometric sequence of 13 points where a₀=a and a₁₃=ar¹²=b and an arithmetic sequence of 16 points where a₀=a and a₁₅=a+15d=b. If we construct these sequences and find that ar⁶=a+7d, what is the length of the interval?

For the geometric sequence, assume that \(\displaystyle a>0\) so let \(\displaystyle r=\sqrt[12]{\frac{b}{a}}\).

For the A.P. let \(\displaystyle \Delta=\dfrac{b-a}{15}\) then \(\displaystyle a_n=a+n\Delta,~n=0,1,\cdots,15\).