Question : Find the average of odd numbers from 11 to 1345
Correct Answer 678
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 11 to 1345
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 11 to 1345 are
11, 13, 15, . . . . 1345
After observing the above list of the odd numbers from 11 to 1345 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 11 to 1345 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 11 to 1345
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 1345
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 1345
= 11 + 1345/2
= 1356/2 = 678
Thus, the average of the odd numbers from 11 to 1345 = 678 Answer
Method (2) to find the average of the odd numbers from 11 to 1345
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 11 to 1345 are
11, 13, 15, . . . . 1345
The odd numbers from 11 to 1345 form an Arithmetic Series in which
The First Term (a) = 11
The Common Difference (d) = 2
And the last term (ℓ) = 1345
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 1345
1345 = 11 + (n – 1) × 2
⇒ 1345 = 11 + 2 n – 2
⇒ 1345 = 11 – 2 + 2 n
⇒ 1345 = 9 + 2 n
After transposing 9 to LHS
⇒ 1345 – 9 = 2 n
⇒ 1336 = 2 n
After rearranging the above expression
⇒ 2 n = 1336
After transposing 2 to RHS
⇒ n = 1336/2
⇒ n = 668
Thus, the number of terms of odd numbers from 11 to 1345 = 668
This means 1345 is the 668th term.
Finding the sum of the given odd numbers from 11 to 1345
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 1345
= 668/2 (11 + 1345)
= 668/2 × 1356
= 668 × 1356/2
= 905808/2 = 452904
Thus, the sum of all terms of the given odd numbers from 11 to 1345 = 452904
And, the total number of terms = 668
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 1345
= 452904/668 = 678
Thus, the average of the given odd numbers from 11 to 1345 = 678 Answer
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