Question:
Find the average of odd numbers from 3 to 1351
Correct Answer
677
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 3 to 1351
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 3 to 1351 are
3, 5, 7, . . . . 1351
After observing the above list of the odd numbers from 3 to 1351 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 3 to 1351 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 3 to 1351
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 1351
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 3 to 1351
= 3 + 1351/2
= 1354/2 = 677
Thus, the average of the odd numbers from 3 to 1351 = 677 Answer
Method (2) to find the average of the odd numbers from 3 to 1351
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 3 to 1351 are
3, 5, 7, . . . . 1351
The odd numbers from 3 to 1351 form an Arithmetic Series in which
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 1351
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 3 to 1351
1351 = 3 + (n – 1) × 2
⇒ 1351 = 3 + 2 n – 2
⇒ 1351 = 3 – 2 + 2 n
⇒ 1351 = 1 + 2 n
After transposing 1 to LHS
⇒ 1351 – 1 = 2 n
⇒ 1350 = 2 n
After rearranging the above expression
⇒ 2 n = 1350
After transposing 2 to RHS
⇒ n = 1350/2
⇒ n = 675
Thus, the number of terms of odd numbers from 3 to 1351 = 675
This means 1351 is the 675th term.
Finding the sum of the given odd numbers from 3 to 1351
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 3 to 1351
= 675/2 (3 + 1351)
= 675/2 × 1354
= 675 × 1354/2
= 913950/2 = 456975
Thus, the sum of all terms of the given odd numbers from 3 to 1351 = 456975
And, the total number of terms = 675
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 3 to 1351
= 456975/675 = 677
Thus, the average of the given odd numbers from 3 to 1351 = 677 Answer
Similar Questions
(1) Find the average of odd numbers from 13 to 1347
(2) Find the average of odd numbers from 5 to 859
(3) Find the average of the first 4114 even numbers.
(4) What is the average of the first 979 even numbers?
(5) Find the average of even numbers from 4 to 1668
(6) Find the average of even numbers from 8 to 60
(7) Find the average of even numbers from 6 to 1388
(8) Find the average of even numbers from 6 to 32
(9) Find the average of even numbers from 12 to 122
(10) What is the average of the first 125 odd numbers?