श्रेणी `1* n+2*(n-1)+3*(n-2)+ ….+(n-1)*2+n*1` का योगफल ज्ञात कीजिए।

lim_ (n rarr oo n rarr oo) (1.n ^ (2) +2 (n-1) ^ (2) +3 (n-2) + ... + n.1 ^ (2)) / ( 1 ^ (3) + 2 ^ (3) + ... n ^ (3))

1+2+3+…….+(n+1)= ((n+1) (n+2))/(2) , n in N .

If n is a non zero rational number then show that 1 + n/2 + (n (n - 1))/(2.4) + (n(n-1)(n - 2))/(2.4.6) + ….. = 1 + n/3 + (n (n + 1))/(3.6) + (n (n + 1) (n + 2))/(3.6.9) + ….

If n is a non zero rational number then show that 1 + n/2 + (n (n - 1))/(2.4) + (n(n-1)(n - 2))/(2.4.6) + ….. = 1 + n/3 + (n (n + 1))/(3.6) + (n (n + 1) (n + 2))/(3.6.9) + ….

lim_ (n rarr oo) [(1 * n + 2 (n-1) + ... + n * 1) / (1 ^ (3) + 2 ^ (3) + ... + n ^ (3) ) +1] ^ (n)

If A=([x,x],[x,x]) then A^(n)(n in N)= 1) ([2^nx^n,2^nx^n],[2^nx^n,2^nx^n]) 2) ([2^(n-1) x^n,2^(n-1) x^n],[2^(n-1) x^n,2^(n-1) x^n]) 3) I 4) ([2^(n) x^(n-1),2^(n) x^(n-1)],[2^(n) x^(n-1),2^(n) x^(n-1)])

Prove by mathematical induction, n.1+(n-1).2+(n-2).3+…+2.(n-1)+1.n = (n(n+1)(n+2))/6 where n in N .

P(n) : 1^(2) + 2^(2) + 3^(2) + .......+ n^(2) = n/6(n+1) (2n+1) n in N is true then 1^(2) +2^(2) +3^(2) + ........ + 10^(2) = .......

If ninNN , then by princuple of mathematical induction prove that, n*1+(n-1)*2+(n-2)*3+ . . . +2*(n-1)+1*n=(1)/(6)n(n+1)(n+2)