Question:
Find the average of odd numbers from 7 to 987
Correct Answer
497
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 7 to 987
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 7 to 987 are
7, 9, 11, . . . . 987
After observing the above list of the odd numbers from 7 to 987 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 7 to 987 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 7 to 987
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 987
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 7 to 987
= 7 + 987/2
= 994/2 = 497
Thus, the average of the odd numbers from 7 to 987 = 497 Answer
Method (2) to find the average of the odd numbers from 7 to 987
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 7 to 987 are
7, 9, 11, . . . . 987
The odd numbers from 7 to 987 form an Arithmetic Series in which
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 987
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 7 to 987
987 = 7 + (n – 1) × 2
⇒ 987 = 7 + 2 n – 2
⇒ 987 = 7 – 2 + 2 n
⇒ 987 = 5 + 2 n
After transposing 5 to LHS
⇒ 987 – 5 = 2 n
⇒ 982 = 2 n
After rearranging the above expression
⇒ 2 n = 982
After transposing 2 to RHS
⇒ n = 982/2
⇒ n = 491
Thus, the number of terms of odd numbers from 7 to 987 = 491
This means 987 is the 491th term.
Finding the sum of the given odd numbers from 7 to 987
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 7 to 987
= 491/2 (7 + 987)
= 491/2 × 994
= 491 × 994/2
= 488054/2 = 244027
Thus, the sum of all terms of the given odd numbers from 7 to 987 = 244027
And, the total number of terms = 491
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 7 to 987
= 244027/491 = 497
Thus, the average of the given odd numbers from 7 to 987 = 497 Answer
Similar Questions
(1) Find the average of even numbers from 10 to 1842
(2) Find the average of odd numbers from 15 to 595
(3) What will be the average of the first 4401 odd numbers?
(4) What will be the average of the first 4041 odd numbers?
(5) Find the average of the first 3255 odd numbers.
(6) Find the average of the first 1153 odd numbers.
(7) Find the average of even numbers from 10 to 928
(8) What is the average of the first 1303 even numbers?
(9) What is the average of the first 350 even numbers?
(10) Find the average of the first 2732 even numbers.