Page 31 - DIGITAL BOOK MUHAMMAD AZWARDI NASUTION

EXAMPLE 9:

An arithmetic series, prove

a. U n+m +U n-m = 2U n

b. S n+3 -3S n+2 + 3S n+1 -S =0
n
c. S n+2 -S =2a+b(2n+1)
n
Proof:

a)a+(n+m-1)b+a+(n-m-1)b

=2a+(n+m-1+n-m-1)b

=2a+(2n-2m-2)b

=2{a+(n-1)b}

=2U n

b)S n+3 -S n+2 -2(S n+2 – S n+1 )+S n+1 -S n

=(U n+3 -U n+2 ) –(U n+2 - U n+1 )=b – b =0

c)S n+2 -S n+1 +S n+1 -S n

=U n+2 + U n+1

= a+(n+2-1)b + a + (n+1-1)b

=2a+(n+1+n)b

=2a+(2n+1)b