Unknown

6.4. Problems 201

n b(n,O) b(n, 1) b(n,2) b(n,3) b(n,J) b(n,5) b(n,6)

0 1

1 1 -1

2 1 -3 2

3 1 -6 11 -6

4 1 -10^35 -50^24

5 1 -15^85 -225^274 -120

6 1 -21^175 -735^1624 -1764^720

7 1 -28^322 -1960^6769 -13132^13068

8 1 -36 546 -4536 22449 -67284 118124

Use this table and the relations developed earlier in this section to find
the sum of the first n kth powers for^1 5 n <^8 and^1 5 k^2 7. Look into
the possibility of determining a formula for the sum in general. Can you
verify the familiar formulae for k = 1,2,3? Assess for iti effectiveness this
approach for getting a general formula.

6.4 Problems

1. (a) If the roots of x3+ ax2+bx+c = 0 are in arithmetic progression,
   prove that 2a3 - 9ub + 27~ = 0.
   (b) If the roots of x3 + ax2 + bx + c = 0 are in geometric progression,
   show that u3c = b3.


2. Given the product p of the sines of the angles of a triangle and the
   product q of their cosines, show that the tangents of the angles are
   the roots of the equation


3. If a, b, c are the roots of the equation

x3 - x2 - x - 1 = 0,

(a) show that a, b, c are all distinct,

(b) show that - b” -c” - c” -a” a” - -b”

b-c + c-u ?- u-b

is an integer for n = 1,2,....

4. It is given that the roots of the equation

17x4 + 36x3 - 14x2 - 4x + 1 = 0

are in harmonic progression. Find these roots.