Question

. The average of 7 consecutive odd number is A. If next 4 and
previous 3 odd numbers to these 7 odd numbers are also
included, then what is the new average of these 14 consecutive
odd numbers?​

Answer 1

Answer:

A+1

Step-by-step explanation:

let assume 7 consecutive odd no.= x,x+2,x+4,x+6,x+8,x+10,x+12

7x+42=7A

x+6=A

if add next four term and previous 3 term then

x+14,x+16,x+18,x+20,x-2,x-4,x-6

14x+98/14=x+7

then new average =A+1

Answer 2

Answer:

The average of 14 consecutive odd numbers if the next 4 and previous 3 odd numbers are added to the given set of 7 odd numbers = A+1

Step-by-step explanation:

Given,

The average of 7 consecutive odd numbers is A

To find,

The average of 14 consecutive odd numbers if the next 4 and previous 3 odd numbers are added to the given set of 7 odd numbers

Recall the concept,

The sum of first 'n' terms of an AP,

$S_n = \frac{n}{2}[2a+(n-1)d]$, -------------------(1)

where 'a'  is the first term  and 'd' is the common difference of the AP

Average = $\frac{sum \ of \ values }{Total \ number \ of \ values}$ ---------------------(2)

Solution:

Let us take the first odd number in the set of 7 consecutive odd numbers be 'a'.

Since the difference between any two odd numbers is 2, we have d = 2

Sum of values = Sum of seven consecutive odd numbers

Substituting the value of the first term as 'a' and common difference d = 2, in equation (1) we get

$S_7 = \frac{7}{2}[2a+(7-1)X2]$

$= \frac{7}{2}[2a+12]$

= 7(a+6)

Sum of values= 7(a+6)

Total number of values = 7

Substituting equation (2) we get,

Average = $\frac{sum \ of \ values }{Total \ number \ of \ values}$ = a+6

Since the average of 7 consecutive odd numbers is A, we have

a+6 = A

a = A-6 ----------------(3)

When  4 odd numbers are added to the next and 3 odd numbers are added previous to the above set, we get

The first term of the above set of 14 numbers is a - 6 and the common difference = 2

Sum to 14 terms of the new set of odd numbers = $S_{14} = \frac{14}{2}[2(a-6) + 13X2]$

= 14(a-6+13)

= 14(a+7)

Substituting the value of 'a' from equation (3) we get

S₁₄ = 14(A-6+7) = 14(A+1)

Sum of values = 14(A+1)

Total number of values = 14

Average = $\frac{14(A+1)}{14}$ = A+1

∴ The new average of these 14 consecutive odd numbers = A+1

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