Question

A manufacturer TV sets produced 600 units in the third year and 700 units in the seventh year. Assuming that the production increases uniformly by a fixed number every year, find the production in the
a) The first year …………… (a)
b) The 10th year……………. (b)
c) 7 years ……………………. (c)
d) Find (c)-(a)-(b)

Answer 1

Hint: As the production increases uniformly by a fixed number every year, the production of the units of TV in the years will be in the form of an arithmetic progression. We can use the formulas for finding the nth term of an A.P. and the sum of n terms of an A.P. to obtain the answer to the question.

Complete step-by-step answer:
It is given that production increases uniformly by a fixed number every year, let’s say by a fixed number d. Therefore, if the no. of units of TV produced in the first year is a, then the units produced in the second year will be a+d, third year will be a+2d and so on.

Therefore, the number of units produced in the years will be an arithmetic progression with initial term a and common difference d. Therefore,

The number of units of TV produced in nth year= a+(n-1)d ………………………………..(1.1)

a)It is given that the number of units of TV produced in third and seventh year is 600 and 700 respectively. Thus, form (1.1),

$a+(3-1)d=600\Rightarrow a+2d=600........(1.2)$

And $a+(7-1)d=600\Rightarrow a+6d=700.........(1.3)$


Subtracting (1.2) from (1.3), we get

$4d=100\Rightarrow d=25...............(1.4)$

Using this value of d in equation (1.2), we get

$a+2\times 25=600\Rightarrow a=550..............(1.5)$


Therefore, the number of units produced in 1st year=550


b)Using equations (1.1), (1.4) and (1.5), we get

No. of units produced in 10th year=$a+(10-1)d=550+9\times 25=775............(1.6)$


c)The sum of n terms of an AP is given by

${{s}_{n}}=\dfrac{n}{2}\left( 2a+(n-1)d \right)$

Taking n=7, and using equation (1.4) and (1.5) to get a and d,


\[The\text{ }production\text{ }in\text{ }7\text{ }years=\dfrac{7}{2}\left( 2\times 550+6\times

25 \right)=4375..............(1.7)\]

d)Thus, from equations (1.5), (1.6) and (1.7), we find that

(c)-(a)-(b)= 4375-775-550 = 3050

Which is the answer to the question.

Note: We should be careful that the nth term of an A.P. is given by \[a+\left( n-1 \right)d\] and not \[a+nd\] because in the first term, d is not added and gets added only from the second term onwards.