Relating values algebraically in exponential function

Question

I'm working on an old problem based on SICP and I'm a bit confused by how to relate the variables of expt-iter in the way that the problem is asking below:

Give an algebraic formula relating the values of the parameters b, n, counter, and product of the expt and exp-iter procedures. (The kind of answer we’re looking for is “the sum of b, n, and counter times product is always equal to 37.”) (HINT: you might want to make a chart of values)

(define (expt b n)   
  (expt-iter b n 1)

(define (expt-iter b counter product)   
  (if (= counter 0)
      product
      (expt-iter b
                (- counter 1)
                (* b product))))

Taking the advice of the hint, I made a chart of values based on $2^8$:

| b | n | a   |
|---|---|-----|
| 2 | 8 | 1   |
| 2 | 7 | 2   |
| 2 | 6 | 4   |
| 2 | 5 | 8   |
| 2 | 4 | 16  |
| 2 | 3 | 32  |
| 2 | 2 | 64  |
| 2 | 1 | 128 |
| 2 | 0 | 256 |

I'm not sure that I am seeing a pattern in the way that the question is asking which would relate b, n, and a.

Answer 1

I don't have my paper copy handy, but it appears that this is the page in question and that your OP has the code directly. The relationship in your chart is

$$a=b^{8-n}$$

Since it calculates an exponent there wouldn't be an additive relationship. But I will note that what they are asking for means you have not labeled your chart correctly. Note that your chart has the heading $n$ and they are asking for the relationship of $counter$. If you label your chart correctly you will find that $n$ is a constant and what you have labeled as $n$ is actually $counter$ and $a$ is $product$ in which case your relationship is

$$\mathrm{product}=b^{n-\mathrm{counter}}$$