PRELIMINARY ALGEBRA
1.16 Express the following in partial fraction form:
(a)2 x^3 − 5 x+1
x^2 − 2 x− 8, (b)x^2 +x− 1
x^2 +x− 2.
1.17 Rearrange the following functions in partial fraction form:
(a)x− 6
x^3 −x^2 +4x− 4, (b)x^3 +3x^2 +x+19
x^4 +10x^2 +9.
1.18 Resolve the following into partial fractions in such a way thatxdoes not appear
in any numerator:
(a)2 x^2 +x+1
(x−1)^2 (x+3), (b)x^2 − 2
x^3 +8x^2 +16x, (c)x^3 −x− 1
(x+3)^3 (x+1).
Binomial expansion1.19 Evaluate those of the following that are defined: (a)^5 C 3 ,(b)^3 C 5 ,(c)−^5 C 3 ,(d)
− (^3) C 5.
1.20 Use a binomial expansion to evaluate 1/
√
4 .2 to five places of decimals, and
compare it with the accurate answer obtained using a calculator.Proof by induction and contradiction1.21 Prove by induction that
∑nr=1r=^12 n(n+1) and∑nr=1r^3 =^14 n^2 (n+1)^2.1.22 Prove by induction that
1+r+r^2 +···+rk+···+rn=1 −rn+1
1 −r.
1.23 Prove that 3^2 n+7, wherenis a non-negative integer, is divisible by 8.
1.24 If a sequence of terms,un, satisfies the recurrence relationun+1=(1−x)un+nx,
withu 1 = 0, show, by induction, that, forn≥1,
un=1
x[nx−1+(1−x)n].1.25 Prove by induction that
∑nr=11
2 rtan(
θ
2 r)
=
1
2 ncot(
θ
2 n)
−cotθ.1.26 The quantitiesaiin this exercise are all positive real numbers.
(a) Show thata 1 a 2 ≤(
a 1 +a 2
2) 2
.
(b) Hence prove, by induction onm,thata 1 a 2 ···ap≤(
a 1 +a 2 +···+ap
p)p
,wherep=2mwithma positive integer. Note that each increase ofmby
unity doubles the number of factors in the product.