Answer:
The two ratios given: 25/15 and 65/56 do not form a proportion.
Step-by-step explanation:
A proportion is formed when two ratios (or fractions) are equal. When you set the two ratios equal to each other:
You can then determine if they are equal by a few different methods. (1) You can cross-multiply the numbers and see if they are equal:
25 x 56 = 16 x 65 or 1040 ≠ 1400
You can also see if the numerators and denominators can be divided to form a common factor:
65 ÷ 25 = 2.6 and 56 ÷ 16 = 3.5
Since neither the products of cross-multiplication or the factors by division are equal, these two ratios do not form a proportion.
Prove:
Using mathemetical induction:
P(n) =
for n=1
P(n) = = 6
It is divisible by 2 and 3
Now, for n=k,
P(k) =
Assuming P(k) is divisible by 2 and 3:
Now, for n=k+1:
P(k+1) =
P(k+1) =
P(k+1) =
Since, we assumed that P(k) is divisible by 2 and 3, therefore, P(k+1) is also
divisible by 2 and 3.
Hence, by mathematical induction, P(n) = is divisible by 2 and 3 for all positive integer n.