Number System-Railway Maths Part VIII

Sum of 1 to 75 natural numbers =
(75 × (75 + 1)) / 2 = (75 × 76) / 2 = 2850
And sum of 1 to 24 natural numbers =
(24 × (24 + 1)) / 2 = 300

Then,
25 + 26 + ……. + 75 = 2850 - 300 = 2550

1/(1×2) + 1/(2×3) + ………. + 1/(99×100)
= 1/2 + 1/6 + 1/12 + ………. + 1/(99×100)
= 99/100 = 0.99

Short Tricks :-
Sum = 1/((n-1)d) × { (1/k) - (1/L) }

Where,
n = no. of terms in denominator
d = difference between the terms
k = first term of first denominator
L = last term of last denominator

Required Sum = 1/((2–1)1) × { (1/1) – (1/100) }
= {99/100} = 0.99

Let numbers are x, y and z
ratio between x, y and z = 6 : 3 : 2
Then, 6 + 3 + 2 = 11 unit
11 unit = 253, 1 unit = 23
So, 2nd number = y = 3 unit = 69

Given, x + y = 10-----(i)
and x² - y² = 60
x² - y² = (x - y) (x + y)
60 = x - y (10),
6 = x- y-----(ii)
Then, From equation (i) and (ii)
2x = 16 ⇒ x = 8
Now, putting the value of x in equation (i) x + y = 10, 8 + y = 10 ⇒ y = 2
Hence, Number are 8 and 2
Alternate method,
We can go through the option and find solution.

Let , Least +ve Integer = x
x² = 5x - 6, x² - 5x + 6 = 0
x = 3 and 2
Hence, least +ve integer = 2

Any number is divisible by 4
if the last 2 digit of the number are divisible by 4
if we put 5 in place of $, the last 2- digit will be 54 and not divisible by 4.

17²⁰⁰/18 = (18 - 1)²⁰⁰/18
Remainder = (-1)²⁰⁰ = 1

Smallest prime number = 2,
greatest prime number = 7
Difference Between the largest and the smallest single-digit prime number
= 7 - 2 = 5

29² = 841
0.000845 - 0.000004 = 0.000841
0.000841 is a perfect square number.
√0.000841 = 0.029