Arithmetic Sequence: Find the 11th Term and Sum of Terms

Q 1.
Answer
An experienced accountant (E) prepares tax return in 16 hours. There-
fore, in 1 hour E prepares: 1
16 tax return.
An novice accountant (N ) prepares tax return in 21 hours. Therefore, in 1
hour N prepares: 1
21 tax return.
If E and N work together, in one hour they will prepare:
1
16 + 1
21 = 37
336 tax return
Suppose, together they require n hours to prepare one complete tax return:
=⇒ n × 37
336 = 1
or,
n = 336
37 ≈ 9.081 hours
Answer: Together they will prepare a tax return in 9.081 hours
Q 2.
Answer
f (x) = 3x + 4
5x + 6
Inverse of f (x): (Dawkins, 2018)
y = 3x + 4
5x + 6
Interchange x and y:
x = 3y + 4
5y + 6
Solve for y:
x(5y + 6) = 3y + 4 =⇒ 5xy − 3y = 4 − 6x =⇒ y = 4 − 6x
5x − 3
1

Replace y with f −1(x):
f −1(x) = 4 − 6x
5x − 3
Composition test:
(f −1 o f )(x) = x
Therefore,
(f −1 o f )(x) = 4 − 6f (x)
5f (x) − 3
= 4 − 6 ( 3x+4
5x+6
)
5 ( 3x+4
5x+6
) − 3
= 20x + 24 − 18x − 24
15x + 20 − 15x − 18 = 2x
2 = 2
hence verified.
Domain of f (x) : It is defined for all real numbers except for the condition
5x + 6 = 0 which occurs at x = −6
5 . Therefore, the domain of f (x) is set of
all real numbers except x = −6
5 : {x, x ∈ R | x 6 = −6
5 }
Domain of f −1(x) : It is defined for all real numbers except for the condition
5x − 3 = 0 which occurs at x = 3
5 . Therefore, the domain of f −1(x) is set of
all real numbers except x = 3
5 : {x, x ∈ R | x 6 = 3
5 }
Range of f (x) : It is the domain of f −1(x), that is the range of f (x) is the
set of all real numbers except y = 3
5
Range of f −1(x) : It is the domain of f (x), that is the range of f (x) is the
set of all real numbers except y = −6
5
Q 3.
Answer
x4 + 18x3 + 71x2 − 18x − 72 = 0
The rational root theorem states that if P (x) is a polynomial with integer
coefficients and if p
q is a root of P (x), then p is a factor of constant term of
P (x) and q is a factor of the leading coefficient of P (x).
2

For the given polynomial P (x) = x4 + 18x3 + 71x2 − 18x − 72, using the ratio-
nal theorem we have, p: a factor of -72 = ±(1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72)
and q: a factor of 1 = ±1
Possible values p
q = ±(1,2,3,4,6,8,9,12,18,24,36,72)
±1 = ±(1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72)
From the possible roots of P (x), the exact roots are selected by synthetic di-
vision of P (x) with each factor. The roots which perfectly divide P (x) are
x = ±1, −6, −12 As P (x) is a polynomial with degree 4, all its roots are
found.
P (x) has a positive leading term with even degree (x4), therefore the ends of
the graph go to +∞ as x goes to +∞ or −∞
Q 4.
Answer
f (x) = 2x2 − 8
x2 + 3x − 10
3