Question:
Find the average of odd numbers from 11 to 987
Correct Answer
499
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 11 to 987
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 11 to 987 are
11, 13, 15, . . . . 987
After observing the above list of the odd numbers from 11 to 987 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 11 to 987 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 11 to 987
The First Term (a) = 11
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 11 to 987
= 11 + 987/2
= 998/2 = 499
Thus, the average of the odd numbers from 11 to 987 = 499 Answer
Method (2) to find the average of the odd numbers from 11 to 987
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 11 to 987 are
11, 13, 15, . . . . 987
The odd numbers from 11 to 987 form an Arithmetic Series in which
The First Term (a) = 11
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 11 to 987
987 = 11 + (n – 1) × 2
⇒ 987 = 11 + 2 n – 2
⇒ 987 = 11 – 2 + 2 n
⇒ 987 = 9 + 2 n
After transposing 9 to LHS
⇒ 987 – 9 = 2 n
⇒ 978 = 2 n
After rearranging the above expression
⇒ 2 n = 978
After transposing 2 to RHS
⇒ n = 978/2
⇒ n = 489
Thus, the number of terms of odd numbers from 11 to 987 = 489
This means 987 is the 489th term.
Finding the sum of the given odd numbers from 11 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 11 to 987
= 489/2 (11 + 987)
= 489/2 × 998
= 489 × 998/2
= 488022/2 = 244011
Thus, the sum of all terms of the given odd numbers from 11 to 987 = 244011
And, the total number of terms = 489
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 11 to 987
= 244011/489 = 499
Thus, the average of the given odd numbers from 11 to 987 = 499 Answer
Similar Questions
(1) Find the average of odd numbers from 13 to 629
(2) What is the average of the first 948 even numbers?
(3) What is the average of the first 728 even numbers?
(4) Find the average of even numbers from 4 to 526
(5) Find the average of the first 3666 even numbers.
(6) Find the average of even numbers from 10 to 436
(7) Find the average of even numbers from 4 to 1478
(8) Find the average of even numbers from 8 to 1342
(9) Find the average of the first 649 odd numbers.
(10) Find the average of the first 4072 even numbers.