model 2 average of consecutive numbers Section-Wise Topic Notes With Detailed Explanation And Example Questions

Answer: (a)

Using Rule 1,

Average of numbers = ${x_1+x_2+…+x_n}/n$

The average will increase by 1.5.

As, ${1+2+3+4+5}/5$ = 3,

${1+2+3+4+5+6+7+8}/8$ = 4.5

⇒ 4.5 - 3 = 1.5

Answer: (c)

Using Rule 1,

Average of numbers = ${x_1+x_2+…+x_n}/n$

x+x+1+x+2+x+3+x+4+x+5+x+ 6 = 26 × 7

⇒ 7x + 21 = 182

⇒ 7x = 182 – 21 = 161

⇒ x = $161/7$ = 23

Shortcut Approach

Fourth number = 26

∴ First number = 23

Answer: (b)

Required average

= ${2+3+5+7+11+13+17+19+23+29}/10$

= $129/10$ = 12.9

Answer: (c)

Formula,

Average of the first n natural odd numbers = $n/2$

Number of odd numbers upto $100/2$ = 50 = required average

Aliter : Using Rule 7,

Odd numbers are 1, 3, 5, ............., 99

Total odd numbers are= 50

∴ Average = 50

Answer: (a)

Using Rule 4,

Average of $1^2,2^2,3^2,4^2…x^2$ = ${(n+1)(2n+1)}/6$

${1^2+2^2+3^2+…+n^2}/n$ = ${(n+1)(2n+1)}/6$

∴ ${1^2+2^2+3^2+…+10^2}/10$ = ${(10+1)(2×10+1)}/6$

= ${11×21}/6$ = 38.5

Answer: (c)

Five consecutive integers are :

x, x + 1, x + 2, x + 3 and x + 4

∴ Their average

= ${x+x+1+x+2+x+3+x+4}/5$

= ${5x+10}/5$ = x+2

New average

= ${(5x+10)+x+5+x+6}/7$

= ${7x+21}/7$= x + 3

Difference = x + 3 – x – 2 = 1