Question

find (50²-49²)+(48²-47²)+(46²-45²)+.......+(2²-1²)​

Answer 1

Answer:

1275

Step-by-step explanation:

METHOD 1

Above series can be broken into two different parts .

Even sum = 2n(n+1)(2n+1)/3

Odd sum = n(2n+1)(2n-1)/3

50²-49²)+(48²-47²)+(46²-45²)+.......+(2²-1²)

=( 50^2 + 48^2 ...+2^2 )-(49^2 + 47^2 ..+1^2)

= 2(25)(26)(50+1)/3 - 25(50+1)(50-1)/3

= 50*26*51/3 - 25*51*49/3

= (66300 - 62475)/3

= 3825/3 = 1275

METHOD 2

(50+49)(50-49) + (48+47)(48-47) + (46+45)(46-45)+.....(2+1)(2-1)

= 99*1 + 95*1 + 91*1 + ....3*1

Tn = 99 + (n-1)(-4)

3 = 99-4n + 4

4n = 100

n = 25

Sn = (25/2)[2(99) + (n-1)d]

= (25/2)[198+(24)(-4)]

= 12.5×(198-96)

= 12.5 × 102

= 1275