Question:
Find the average of odd numbers from 15 to 343
Correct Answer
179
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 343
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 343 are
15, 17, 19, . . . . 343
After observing the above list of the odd numbers from 15 to 343 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 343 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 343
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 343
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 15 to 343
= 15 + 343/2
= 358/2 = 179
Thus, the average of the odd numbers from 15 to 343 = 179 Answer
Method (2) to find the average of the odd numbers from 15 to 343
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 343 are
15, 17, 19, . . . . 343
The odd numbers from 15 to 343 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 343
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 15 to 343
343 = 15 + (n – 1) × 2
⇒ 343 = 15 + 2 n – 2
⇒ 343 = 15 – 2 + 2 n
⇒ 343 = 13 + 2 n
After transposing 13 to LHS
⇒ 343 – 13 = 2 n
⇒ 330 = 2 n
After rearranging the above expression
⇒ 2 n = 330
After transposing 2 to RHS
⇒ n = 330/2
⇒ n = 165
Thus, the number of terms of odd numbers from 15 to 343 = 165
This means 343 is the 165th term.
Finding the sum of the given odd numbers from 15 to 343
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 15 to 343
= 165/2 (15 + 343)
= 165/2 × 358
= 165 × 358/2
= 59070/2 = 29535
Thus, the sum of all terms of the given odd numbers from 15 to 343 = 29535
And, the total number of terms = 165
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 15 to 343
= 29535/165 = 179
Thus, the average of the given odd numbers from 15 to 343 = 179 Answer
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