Question

(c) The total number of one centimeter lines in the first n diagrams is given by the expression

23n3 + fn2 + gn.



(i) Use n = 1 in this expression to show that f + g =103. [1]



(ii) Use n = 2 in this expression to show that 4f + 2g = 323. [2]



(iii) Find the values of f and g​

Answer 1

I) f + g = 10/3

II) 4f + 2g = 20/3

III) f = 2 and g = 4/3

Explanation:

From the chart,

P = 25

q = 40

The total number of one centimeter lines in the first n diagrams is given by the expression

2/3n^3 + fn^2 + gn.

When n = 1, the total number of line = 4. So,

2/3(1)^3 + f(1)^2 + g(1) = 4

2/3 + f + g = 4

Make f+g the subject of formula

f + g = 4 - 2/3

f + g = (12 - 2)/3

f + g = 10/3 ......(1)

When n = 2

Total number of line = 12

2/3(2)^3 + f(2)^2 + g(2) = 12

2/3×8 + 4f + 2g = 12

16/3 + 4f + 2g = 12

4f + 2g = 12 - 16/3

4f + 2g = (36 - 16)/3

4f + 2g = 20/3 ......(2)

(iii) To find the values of f and g, solve equation 1 and 2 simultaneously

f + g = 10/3 × 2

4f + 2g = 32/3

2f + 2g = 20/3

4f + 2g = 32/3

- 2f = - 12/3

f = 12/6

f = 2

Substitutes f in equation 1

f + g = 10/3

2 + g = 10/3

g = 10/3 - 2

g = (10 - 6)/3

g = 4/3