n = 3 (1+2+3)/3 = 6/3 = 2n = 7 (2+3+4+5+6+7+8)/7 = 35/7 = 5

n = 2 (1+2)/2 = 1.5 n=4 (3+4+5+6)/4 =4.5

n = 2 (1.2)/2! = 1 n=5 (3.4.5.6.7)/5! =4.5

ODD Divisible

EVEN NOT Divisible

Multiple of N ± Multiple of N = Multiple of N

Multiple of N ± Non-Multiple of N = Non-Multiple of N

Non Multiple of N + Non Multiple of N = Multiple or Non-multiple of N

(X^3-Y^3) = (X-Y)(X^2+XY+Y^2)

xn - yn = ( x - y ) (xn-1 + xn-2y + ... + xyn-2 + yn-1 )

Because (x2 – y2 ) = (x-y)(x+y)X4-y4 =(x2-y2)(x2+y2) = (x+y)(x-y)(x2+y2)

(a + b)3 = a3 + 3a2b + 3ab2 + b3 (a - b)3 = a3 - 3a2b + 3 ab2 - b3

a3 - b3 = (a - b)3 + 3 a b (a - b)

a3 - b3 = (a - b) (a2 + a b + b2)

a3 + b3 = (a + b)3 - 3 a b (a + b)

a3 + b3 = (a + b) (a2 - a b + b2)

a b = ((a + b) / 2)2 - ((a - b) / 2)2

2.25

2.45

2.85

Inequalities

Inequality Flips No-Flip

Multiply by –ve number Flips

Taking Reciprocal Flip (When both numbers have same sign)

No-flip( when they have different sign)

Squaring

Addition, Substraction, Square root etc does not change inequality

No-Flip

Eg -2 > -3 => -1/2 < -1/3

3 > 2 => 1/3 < 1/2

If the signs are same then

Reciprocal will flip the inequality

If the signs are different then

Reciprocal will not flip the inequality

Squaring will flip the ineq if both numbers are negative (both side multiplied by –ve number)

Operation Mean Median Mode Range StandardDev

ai ± k(increased decreased by k)

Mean ± k Median ± k

Mode ± k Does Not Change

Does Not Change

ai × k Mean × k Median × k

Mode × k Range × k SD × k

ai / k Mean / k Median / k

Mode / k Range / k SD / k

Mean = Average (Sum/# of elements)Median = middle number (consec numbers median = mean, n=odd=> median=int)Mode = most frequentRange = max - minSD = Sqrt(Variance) Variance = Ʃ (mean – xi)2 / # of terms n

Statistics