Question

Help #3 11/05/2017
Please help me do this question.

Answer 1

(5)

let P(x ) = ax³ + bx² + cx + d

question says
P(i) = 1/( i + 1) where i = { 1, 2, 3 , 4 }

P(1) = 1/(1 + 1) = 1/2
P(2) = 1/(2 + 1) = 1/3
P(3) = 1/(3 +1) = 1/4
P(4) = 1/(4 + 1) = 1/5

but we assume
P(x) = ax³ + bx² + cx + d
P(1) = a + b + c + d
P(2) = 8a + 4b + 2c +d
P(3)= 27a + 9b +3c + d
P(4)= 64a + 16b + 4c + d

so,

P(1) = a + b + c + d = 1/2 ---------(1)
P(2) = 8a +4b + 2c +d = 1/3 ------(2)
P(3)= 27a + 9b + 3c + d = 1/4 -----(3)
P(4) = 64a +16b + 4c +d = 1/5 -------(4)

equations (2) - (1)

7a + 3b + c = -1/6 -------(5)

eqn (3) -(2)

19a + 5b + c = -1/12 ------(6)

eqn , (4) - (3)
37a + 7b + c = -1/20 --------(7)


now , eqn (6) - (5)
12a + 2b = 1/12 ---------(8)

eqn ( 7) - (6)

18a + 2b = 1/30 ------(9)

solve equations (8) and (9)

6a = 1/30 - 1/12 = -1/20

a = -1/120 put in equation (8)

-1/10+ 2b = 1/12
2b = 1/12 + 1/10 = 11/60
b = 11/120

put a and b value in equation (5)

7( -1/120) +3( 11/120) + c = -1/6

( -7 +33)/120 + c = -1/6
c = (-20-26)/120 = -46/120


put a, b, c , in equation (1)

d = 1/2 - ( -1/120 + 11/120 -46/120 )
= ( 60 +1 -11 +46)/120
= 96/120


so,

P(x ) = ( -x³ + 11x² -46x + 96)/120

now,

P(5) = ( -125 +11×25 -46×5+96)/120
=(-125 + 275 -230 +96)/120
= ( 16)/120
= 2/15

Answer 2

Hope this will help u... :-)