Question

please do both of them and in induction question kindly write base step as well as inductive step

Let S : R+ Z defined by $(x) = x - 2. Prove that is a one-to-one correspondence Q2 Use induction to prove that E4;* = n* + 2n" + nwhere n 20.

Answer 1

fcal = x²_2 TO 3 - 2 both sideo) Let fi lR Z defined by One-to-one correspondance means given function is one one and onto cheek f is one-one we will show if free, ) = f(x2) for HH, H2 EUR then x = x2 consider fox, I = feres) - 243-2 = re, (cahcle 2 x3 = 3 H, = ( since clube root of positive and negative number is again positive and negative respectively and it is equal.) Next f is onto we will check & igang ) CIR sit f(x)=y so, consider fors=y for every y Ere there exist some +3_2=4 +3= y +2 x=3 y+2 but if we take y=-4EZ then H=35-2 & 1R Hence not onto f it not one-to one correspondance Hence 3 2 = nu+an²+n' for nz Given E4J J=0 Basic step 3+ (1) 2 =4 First we will prove result for n=1 R.H.S. = (1) + 2(1) 2.4.5=²4j3 e 26003+4(123=4 J=0 Hence result is true for n=1.