Question

the sum of four consecutive numbers in an Arithmetic progression. is 32 and the ratio of the product of the first and last term to the product of two middle terms is 7:15. find the numbers ​

Answer 1

The numbers are 2, 6, 10, 14.

That's because $\frac{a_1+a_4}{2} \cdot 4 = 32$ => $a_1+a_4=16$.

And, $\frac{a_1a_4}{a_2a_3}=\frac{7}{15}$, so right now, our $a_1=1, a_2=3, a_3=5, a_4=7$. Now, scaling these 4 numbers up by a factor of 2 yields their sum to be 32, and the ratio to be $\frac{7}{15}$, so we can be sure that $\boxed{a_1=2, a_3=6, a_3=10, a_4=16}$

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