Question

2 a factory makes custom sports cars at an increasing rate. In the first month only one car is made, in the second month two cars are made, and so on, with n cars made in the nth month.
a. Give recurrence relation for the number of cars produced in the first n months by this factory. (4)
b. How many cars are produced in the first year? (4) nov/dec 16 ap

Answer 1

Answer:

a)

$f(x)=\frac{x}{2}(1+n)$  , where x=1,2,...,n

b)

78 cars

Step-by-step explanation:

a)

The number of custom sports cars made is a function of the number of month since the start of the production.

Let x denote the number of cars produced by the factory by a given month since production,

Then we know that;

in the month 1, only 1 car is made,

in month 2, 2 cars are made,

so by the end of month 2 since the start of production, (1+2=3) care are made.

month 1= 1

month 2= 1+2

....

month n= 1+2+...+n

We can see that the number of cars made by the end of the nth month is the sum of an arithmetic progression up to n with an increament rate of 1 beginning with 1.

$sum=\frac{n}{2} *(a+l)$

where a is the first term, in our case it is 1.

          l is the last term, in our case it is n

          n is the number of terms, which is n

Therefore;

$f(x)=\frac{x}{2}(1+n)$  , where x=1,2,...,n

b)

The number cars produced in one year;

1 year consists of 12 months;

using the formula above;

$f(12)=\frac{12}{2}*(1+12)\\ =78$

78 cars