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# After being set up, a pen factory manufactured 16000 pens in the fifth year and 20500 pens in the eighth year. Assuming that production increases uniformly by a fixed number every year find:(i) The number of pens manufactured in the first year.(ii) The total production in 10 year.

Last updated date: 15th Jun 2024
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Hint: A sequence is a list of items/objects which have been arranged in a sequential way.
A series can be highly generalized as the sum of all the terms in a sequence however, there has to be a definite relationship between all the terms of the sequence.
Arithmetic sequence:
A sequence in which every term is created by adding or subtracting a definite number to the preceding number is an arithmetic sequence.
Sequence - $a,\,\,a + d\,,\,\,a + 2d.....$
- $a\left( {n - 1} \right)d.....$
General term (nth term) - ${a_n} = a + (n - 1)d$

Complete step by step solution:
Let, no. of pens in the first year be ‘a’.
And uniform increase be ‘d’
From the given statement:
We have,
$a{}_5 = a + \left( {5 - 1} \right)d$
$a + 4d = 16000$ ______ (1).
By using the formula $\left( {a + \left( {n - 1} \right)d} \right)$
Where n $= 5$year
And in ${18^{th}}$year factory manufactured $20500$pens.
Now ${a_{18}} = a + \left( {18 - 1} \right)d$
$a + 17d = 20500$ _______ (2).
By subtracting the equation (1) from equation (2).
$a + 17d - a - 4d = 20500 - 16000$
$13d = 4500$
$d = \dfrac{{4500}}{{13}}$
$d = 300$.
By putting the volume of ‘d’ in equation (1)
We get,
$a + 4 \times 300 = 16000$
$a + 1200 = 16000$
$a = 16000 - 1200$
$a = 14800$.
(ii) The total production in $10$years
For $10$have:
$a + 9d$ ________ (3)
By putting the value of ‘a’
$14800 + 9 \times 300$
$= 17500$.
In $10$years we having two leap years so that total production is $10$is equal to:
$= \dfrac{7}{2} \times \left( {14800 + 17500} \right)$
$= 7 \times 16150$
$= 113050$.
Hence, the number of pens manufactured in the first year $14800$ pens.
And the total production in $10$years $113050$pens.

Note: In a question if it is given that the quantity is uniformly increasing, this means to say that the sequence is an Arithmetic Progression(A.P)