model 4 find nth average from 1st & last number Section-Wise Topic Notes With Detailed Explanation And Example Questions

Answer: (d)

Let three numbers be a,b and c respectively.

∴ a + b + c = 16 × 3 = 48 ---(i)

b + c + 20 = 15 × 3 = 45

⇒ b + c = 45 – 20 = 25 ---(ii)

By equation (i) - (ii),

a = 48 – 25 = 23

Aliter : Using Rule 16,

Here, n = 3, F = 16, L = 15

l = 20, f= ?

f – l= n (F – L)

f – 20= 3 (16 – 15)

f = 3 + 20

f = 23

Answer: (a)

Seventh observation

= 65 × 7 + 7 × 75 – 13 × 70

= 455 + 525 – 910

= 980 – 910 = 70

Aliter :

Here, n = 13, x = 70

m =7, y = 65

m = 7, z = 75

Seventh result = m (y + z) – nx

= 7 (65 + 75) – 13 × 70

= 7 × (140) – 910

= 980 – 910 = 70

Answer: (a)

Using Rule 1,

Average of two or more numbers/quantities is called the mean of these numbers, which is given by

$\text"Average(A)" = \text"Sum of observation / quantities"/\text"No of observation / quantities"$

∴ S = A × n

Last number = Sum of 20 numbers

– sum of first 12 numbers – sum of next 7 numbers

= 20 × 12 – 11 × 12 – 7 × 10

= 240 – 132 – 70 = 38

Answer: (c)

Using Rule 1,

Sum of 8 numbers = 20 × 8 = 160

Sum of the first two numbers =$31/2$ ×2 = 31

Sum of next three numbers = $64/3$×3=64

Sum of the remaining three numbers = 160 – (31 + 64) = 160 – 95 = 65

Let 6th number = x

∴ 7th number = x + 4,

8th number = x + 7

⇒ x + x + 4 + x + 7 = 65

⇒ 3x = 65 – 11

⇒ x = $54/3$ =18

∴ Eighth number = 18 + 7 = 25

Answer: (c)

Using Rule 1,

The sixth number

= 9 × 50 – 5 × 54 – 3 × 52

= 450 – 270 – 156 = 24

Answer: (a)

Sum of new numbers

= na + (2 + 4 + 8 + 16 ..... to n terms)

Now, S = 2 + 4 + 8 + 16 + ..... to n terms

Here, a = first term = 2

r = common ratio = $4/2$ = 2

It is a geometric series.

∴ S =${a({r^n} - 1)}/{r-1}$= ${2({2^n} - 1)}/{2-1}$

= 2 ($2^n$ –1)

∴ Required average

= ${na +2({2^n}-1)}/n$

a +${2({2^n}-1)}/n$