Question : Find the average of odd numbers from 5 to 1121
Correct Answer 563
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 5 to 1121
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 5 to 1121 are
5, 7, 9, . . . . 1121
After observing the above list of the odd numbers from 5 to 1121 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 5 to 1121 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 5 to 1121
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 1121
The average of the numbers forming an Arithmetic Series
= The first term (a) + The last term (ℓ)/2
⇒ The average of numbers forming an Arithmetic Series = a + ℓ/2
Thus, the average of the odd numbers from 5 to 1121
= 5 + 1121/2
= 1126/2 = 563
Thus, the average of the odd numbers from 5 to 1121 = 563 Answer
Method (2) to find the average of the odd numbers from 5 to 1121
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 5 to 1121 are
5, 7, 9, . . . . 1121
The odd numbers from 5 to 1121 form an Arithmetic Series in which
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 1121
The Average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, to find the average of the given numbers, first, we need to find their sum and the total number of given numbers
Finding the number of terms
For an Arithmetic Series, the nth term
an = a + (n – 1) d
Where
a = First term
d = Common difference
n = number of terms
an = nth term
Thus, for the given series of the odd numbers from 5 to 1121
1121 = 5 + (n – 1) × 2
⇒ 1121 = 5 + 2 n – 2
⇒ 1121 = 5 – 2 + 2 n
⇒ 1121 = 3 + 2 n
After transposing 3 to LHS
⇒ 1121 – 3 = 2 n
⇒ 1118 = 2 n
After rearranging the above expression
⇒ 2 n = 1118
After transposing 2 to RHS
⇒ n = 1118/2
⇒ n = 559
Thus, the number of terms of odd numbers from 5 to 1121 = 559
This means 1121 is the 559th term.
Finding the sum of the given odd numbers from 5 to 1121
The sum of all terms (S) in an Arithmetic Series
= n/2 (a + ℓ)
Where, n = number of terms
a = First term
And, ℓ = Last term
Thus, the sum of all terms (S) of the given odd numbers from 5 to 1121
= 559/2 (5 + 1121)
= 559/2 × 1126
= 559 × 1126/2
= 629434/2 = 314717
Thus, the sum of all terms of the given odd numbers from 5 to 1121 = 314717
And, the total number of terms = 559
Since, the average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, the average of the given odd numbers from 5 to 1121
= 314717/559 = 563
Thus, the average of the given odd numbers from 5 to 1121 = 563 Answer
Similar Questions
(1) Find the average of even numbers from 10 to 1206
(2) Find the average of even numbers from 4 to 1728
(3) Find the average of the first 509 odd numbers.
(4) Find the average of even numbers from 8 to 42
(5) Find the average of even numbers from 12 to 948
(6) Find the average of odd numbers from 3 to 959
(7) Find the average of even numbers from 4 to 1684
(8) What is the average of the first 1281 even numbers?
(9) Find the average of odd numbers from 15 to 205
(10) Find the average of the first 363 odd numbers.