SERIES AND MATHEMATICAL INDUCTION 11

Th.. We will prove that S, = n(n + 1)/2 for all values of n,

that is, that the series of statements

B =20 % 213
e w B W
Sa = n(n + 1)/2

are all tru.
The first of these statements is evidently tru.
Suppose that the kth is tru, that is, that

Sy = k=12
Then Sia = S+ i1 = k(e + 1)/2 +:(R+ 1)
= (k+ 1)(k + 2)/2
Thus Ses=Abhtitbad

So the kth statement in the series implies the 2 + 1th.

Hence the statements are all tru.

This proves that the sum of the first n natural numbers is
n(n + 1)/2.

31. Next consider the series of even numbers

2’4y 6: 87 5P 2y

whose nth term is 27.

Here S, =2+44+6+ - +2n=2(14+2+4+34--- +n)

But14+24+3+ - +n=mnn+1)/2

Hence 2(1 +2 43+ -+ + n) = n(n + 1)

Th. Therfor the sum of the first n even numbers starting with
21is n(n + 1).

32. Consider the odd numbers 1, 3, 5, 7, ---, the nth of
which is 2z — 1.
Here S, 14+34+54+---4+@2n—1)

2= %@ —1)4(6 1)+ vk ln =11)
2+44+64+---+2n)

—(1+4+14+1+ .- ntimes)
nin+1) —n =n

i