Please help

[Sarah123]

Prove, that if positive real numbers a1, a2, a3 form three consecutive terms of a geometric sequence, then their decadic logarithms log a1, log a2, log a3 form three consecutive terms of an arithmetic sequence.

 

[topsquark]

If the a's are in geometric progression then they follow the formula $$a_n = a_0 r^n$$. So your three a's will be
$$a_k = a_0 r^k$$, $$a_{k + 1} = a_0 r^{k + 1} = r a_k$$ and $$a_0 r^{k + 2} = r^2 a_k$$ for some k.

What are the logarithms of these numbers?

-Dan

 

[Sarah123]

If the a's are in geometric progression then they follow the formula $$a_n = a_0 r^n$$. So your three a's will be
$$a_k = a_0 r^k$$, $$a_{k + 1} = a_0 r^{k + 1} = r a_k$$ and $$a_0 r^{k + 2} = r^2 a_k$$ for some k.

What are the logarithms of these numbers?

-Dan

Thank you. I dont know the logarithms of the numbers, they were not specified in the assignment.

 

[pka]

Thank you. I dont know the logarithms of the numbers, they were not specified in the assignment.

Decadic logarithms are $\log_{10}(x)$ called common logarithms.

 

[Dr.Peterson]

I dont know the logarithms of the numbers, they were not specified in the assignment.

You don't need to know any numerical values. Do it symbolically, which is what topsquark suggested.

If a term is [MATH]a_k = a_0 r^k[/MATH], for example, then [MATH]log(a_k) = log(a_0 r^k) = log(a_0) + log(r^k) = log(a_0) + k log(r)[/MATH]. Since [MATH]a_0[/MATH] and [MATH]r[/MATH] are constants, we can think of this as [MATH]log(a_k) = b_0 + k d[/MATH], where [MATH]b_0 = log(a_0)[/MATH] and [MATH]d = log(r)[/MATH]. Does that look familiar?

Or, you could do it a little differently. If the numbers form a geometric sequence, then their ratios are the same. For the logs to form an arithmetic sequence, their differences have to be the same. What is the differences of the logs of two numbers whose ratio is r (as an expression in terms of r)?

 

[topsquark]

Thank you. I dont know the logarithms of the numbers, they were not specified in the assignment.

Let me give an example of what's needed here. $$log ( x y^3) = log(x) + log(y^3) = log(x) + 3 log(y)$$.

Do that for all three values and see if they fit into an arithmetic series.

You don't need the actual numbers.

-Dan