Question

Could you help me with 11,12,and 13 please ?

Thanks in advance,

Massimo Ulto

8. If you have had calculus, prove the power rule for positive exponents. Specifically, prove that for every positive integer n, (x") = na"-1. (Hint: Use induction on n and the Product rule, writing " = 9. Prove that for every positive integer n n(n + 1 + 2 + ... +n= 10. Prove that for every positive integer n. 12 +22+... 2 n(n + 1)(2n + 1) 11. Prove for every positive integer n. 19 +23+ 33 + ... 3 /2n = 1) 12. Prove that for every positive integer 2 1.1! + 2 - 2!+---+ n-n!-(n + 1 13. Prove that n < 25. for every positive integer 14. Prove that 11" - 4" is divisible by 7. for all 2 0. Redo the prool using congruences instead of induction,

Answer 1

A-11 for n= 1 2 +2 3+---+n? = (n (n+1)/ ) use Induction Method - *** 13 = 44.() = 1.1 Now , assume the formula is true for n=k ie, +23+ -- +k%= (klictum Check for n=k+1 1, 2+2+3*+ KB +(+13) = (k~***** (*)° _ (+239 f*** **s] – (k+2)2 [ *** *16*4] Ķ = (k+}*(K4232 = [(*2) (2 that means by 1+28+3='— n3 = (nenaz) ) for every positive integer no 4 Rk+1) (K+2) 2 Induction Q-12 1.11 +221 +-- - +nini =(n+1) 1 - 1 We can write hine - (n+1 – 1) n = (n+1) ni - ni = (n+1) - for n=1 1.1! = 2! – 1! 2-2! = 3! - 2! 3:31 -41 - 31 nine - (n+1)! – ni adding vertically 1.11 +2.21 + nini = (n+1) -

13 for every positive integer 2" > n Induction method for n= 2 1 1 > 1 which true. is Assume it is true for nam je, am>m We will show that it is also true n=m+1. Now, 2m >m multiply by a e both sides. 2m2 > 2m . o mtl mtm >m+1 ie, antly m41 induction Tan>n for every the integer n. that is