Convert the following products into factorials: (iii) `(n + 1) (n +2) (n + 3).... (2n)` (iv) `1.3.5 . 7 . 9.... (2n-1)`

Convert the following products into factorial: 1.3.5.7.9(2n-1)

Convert the following products into factorial: (n+1)(n+2)(n+3)(2n)

the continued product 2.6.10.14... to n factors is equal to 2np_(n)(b)^(2n)C_(n)(n+1)(n+2)(n+3)...(n+n)2^(n)*(1.3.5......2n-1)

Write the first four terms of each of the following sequence whose nth terms are (i) 2^(n) (ii) n/(n+1) (iii) n^(2)-16 (iv) (3^(n))/(2^(n)+1) (v) (n+4)/(n+1) (vi) log(1+1/n)

lim_ (n rarr oo) (2.5 ^ (n + 1) -3.7 ^ (n + 1)) / (2.5 ^ (n) + 3.7 ^ (n)) =

lim_ (n rarr oo) (7.3 ^ (n + 1) + 3.5 ^ (n + 1)) / (5.3 ^ (n) -9.5 ^ (n))

Prove the following by using the principle of mathematical induction for all n in Nvdots1^(2)+3^(2)+5^(2)+...+(2n-1)^(2)=(n(2n-1)(2n+1))/(3)

lim_ (n rarr oo) (5.2 ^ (n + 1) + 2.3 ^ (n + 1)) / (3.2 ^ (n) -7.3 ^ (n))