Question

Find the middle term of the sequence formed by all three digit numbers which leave a remainder $ 3. $ When divided by $ 4. $ Also, find the sum of all numbers on both sides of the middle term separately.

Answer 1

Hint: An Arithmetic Progression (AP) is the sequence of numbers in which the difference of two successive numbers is always constant.
The standard formula for Arithmetic Progression is – $ {a_n} = a + (n - 1)d $
Where
 $ {a_n} = $ nth term in the AP
 $ a = $ First term of AP
 $ d = $ Common difference in the series
 $ n = $ Number of terms in the AP
Here, first of all will get the required numbers which leave a remainder $ 3. $ When divided by $ 4. $ and then total number of numbers in the series and then the middle term.

Complete step-by-step answer:
Find the three digit numbers which leaves a remainder $ 3 $ when divided by $ 4 $
Try diving numbers starting from $ 100 $ to $ 999 $ and get the required numbers.
The resultant numbers are –
 $ 103,107,111,....999 $
Now, we will find the total number of numbers in the above series.
Here,
 $
  a = 103 \\
  d = 107 - 103 = 4 \\
  {a_n} = 999 \;
  $
Place the above values in the equation - $ {a_n} = a + (n - 1)d $
 $ \Rightarrow 999 = 103 + (n - 1)4 $
Simplify the above equation –
 $
   \Rightarrow 999 - 103 = (n - 1)4 \\
   \Rightarrow 896 = (n - 1)4 \;
  $
It can be re-written as –
 $ \Rightarrow (n - 1)4 = 896 $
 $
   \Rightarrow (n - 1) = \dfrac{{896}}{4} \\
   \Rightarrow (n - 1) = 224 \;
  $
Make “n” the subject. When any term is moved from one side to another, then the sign also changes. Positive terms become negative and vice-versa.
 $
   \Rightarrow n = 224 + 1 \\
   \Rightarrow n = 225 \;
  $
Since the total numbers are odd, middle term can be calculated as –
 $ M = \dfrac{{n + 1}}{2} $
 $
   \Rightarrow M = \dfrac{{225 + 1}}{2} \\
   \Rightarrow M = \dfrac{{226}}{2} \\
   \Rightarrow M = 113 \;
  $
Now, find the middle term using $ {a_n} = a + (n - 1)d $
Where $ n = 113 $
 $ \Rightarrow {a_n} = 103 + (113 - 1)4 $
Simplify the above equation –
 $
   \Rightarrow {a_n} = 103 + (112)4 \\
   \Rightarrow {a_n} = 103 + 448 \\
   \Rightarrow {a_n} = 551 \;
  $
Now, sum of all the terms before middle term is
 $ {{\text{S}}_{n{\text{ }}}} = \dfrac{n}{2}[2a + (n - 1)d] $
Here $ n = 112 $ place the values
 $ \Rightarrow {S_n} = \dfrac{{112}}{2}[2(103) + (112 - 1)4] $
Simplify the above equation –
 $
   \Rightarrow {S_n} = \dfrac{{112}}{2}[206 + (111)4] \\
   \Rightarrow {S_n} = 56[206 + 444] \\
   \Rightarrow {S_n} = 56[650] \\
   \Rightarrow {S_n} = 36,400\;{\text{ }}....{\text{ (A)}} \;
  $
Similarly, sum of all the terms is
 $ {{\text{S}}_{n{\text{ }}}} = \dfrac{n}{2}[2a + (n - 1)d] $
Here $ n = 225 $ place the values
 $ \Rightarrow {S_n} = \dfrac{{225}}{2}[2(103) + (225 - 1)4] $
Simplify the above equation –
 $
   \Rightarrow {S_n} = \dfrac{{225}}{2}[206 + (224)4] \\
   \Rightarrow {S_n} = \dfrac{{225}}{2}[206 + 896] \\
   \Rightarrow {S_n} = \dfrac{{225}}{2}[1102] \\
   \Rightarrow {S_n} = 225[551] \\
   \Rightarrow {S_n} = 123975\;{\text{ }}....{\text{ (B)}} \;
  $
Sum of all the terms after middle terms can be given as –
From the Summation of all the terms subtract the terms before middle term and the middle term
 $ = 123975 - 36400 - 551 $
Simplify the above equation –
\[ = 87024\]
So, the correct answer is “\[ 87024\] ”.

Note: Know the difference between the arithmetic and geometric progression. In arithmetic progression, the difference between the numbers are constant in the series whereas, the geometric progression is the sequence in which the succeeding element is obtained by multiplying the preceding number by the constant and the same continues for the series. The ratio between the two remains the same.