sequence and series.

[suraj]

If the sequence a(n) satisfies the relation a(n + 1) = (a(n) − 1)square + a(n), and a(1) = 3, what is the largest common factor of a(15) and a(45)?


is it ok for you sir.?

[ravihanda]

I am not very sure but I am guessing 1.
Look at the first few terms.
a(1) = 3
a(2) = 3 + 2^2 = 7 = 1 + 2 + 2^2
a(3) = 7 + 6^2 = 43 = 1 + 6 + 6^2
a(4) = 43 + 42^2. = 1 + 42 + 42^2

Looking at it I can speculate (not confirm) that the HCF will be 1.

[grimreaper]

If the sequence a(n) satisfies the relation a(n + 1) = (a(n) − 1)square + a(n), and a(1) = 3, what is the largest common factor of a(15) and a(45)?


is it ok for you sir.?

hcf would be one...

here is the solution though.... of the Q.... and am sure you took it from the same source
cheers

"We have an + 1 = (an – 1)2 + an = an2 – an + 1, or an + 1 – 1 = an(an – 1)

We can further write (an – 1) on the RHS as (an – 1) = an − 1(an − 1 – 1)

Proceeding in this fashion, we get an + 1 – 1 = an(an – 1) = an an − 1(an − 1 – 1) = … = an an − 1 an − 2 … a1(a1 – 1)

This gives an + 1 = 1 + an an − 1 an − 2 … a1(a1 – 1)

Putting n = 44, we get a45 = 1 + 2 × a44 a43 … a15 … a1 = 1 + Ka15, where K is an integer.

Since a45 is one more than an integer multiple of a15, they must be relatively prime, and the largest common factor they can have is 1.

Hence, option 5."