The average of 5 consecutive odd no. Is k. After adding 3 next consecutive odd no. new average. .new average will increase by old average?

Question

Answer

Let n be any odd integer. Then the next consecutive integers are (n + 2), (n + 4), ... so on.
k = (n + (n + 2) + (n + 4) ... + (n + 8))/5
k = n + 4
The next three consecutive odd numbers will be (n + 10), (n + 12), (n + 14)
new total = n + (n + 2) + ... + (n + 14)
= 8n + 56
new average = (8n + 56)/8 = n + 7
=> k + 3

Answer

Answer

Problem Statement:

Find out whether adding 3 consecutive odd numbers to the average of 5 consecutive odd numbers increases the average by the old average or not.

Solution:

Let's assume the first odd number to be 'x', so the next 4 consecutive odd numbers will be (x+2), (x+4), (x+6), and (x+8).

The sum of 5 odd numbers will be:

x + (x+2) + (x+4) + (x+6) + (x+8) = 5x + 20

The average of 5 odd numbers will be:

Average = (5x + 20) / 5 = x + 4

So, the average of 5 consecutive odd numbers is 'x+4'.

Now, let's add the next 3 consecutive odd numbers to the previous 5 odd numbers, which will be (x+10), (x+12), and (x+14).

The sum of 8 odd numbers will be:

x + (x+2) + (x+4) + (x+6) + (x+8) + (x+10) + (x+12) + (x+14) = 8x + 56

The average of 8 odd numbers will be:

New Average = (8x + 56) / 8 = x + 7

Now, we need to check whether the new average is greater than the old average or not.

Let's subtract the old average from the new average:

New Average - Old Average = (x+7) - (x+4) = 3

As we can see that the difference between the new average and the old average is 3. So, the new average is greater than the old average by 3.

Hence, the addition of 3 consecutive odd numbers to the average of 5 consecutive odd numbers increases the average by the old average.

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