Question

sum of n terms of (1) ,(1+2,+(1+2+3),.......is​

Answer 1

Answer:

In the series 1+(1+2)+(1+2+3)....., the nth term is the sum of numbers from 1 to n.

please refer to the attachment

So the nth term is n(n+1)/2.

So we need to find the sum of a series whose nth term is n(n+1)/2.

Answer 2

Step-by-step explanation:

Let us Consider that number of Birds are = \sf x^2x

2

The number of Birds moving about lotus plants = \sf\dfrac{x^2}{4}

4

x

2

Number of Birds along = \sf\dfrac{x^2}{9} + \dfrac{x^2}{4}

9

x

2

+

4

x

2

And, the Number of Birds who are moving on a Hill = \sf \: 7 \sqrt{x^2}7

x

2

:\implies\sf 7x:⟹7x

Remain Birds = 56

As Per Given Question -

:\implies\sf \dfrac{x^2}{4} + \dfrac{x^2}{4} + \dfrac{x^2}{9} + 7x + 56 = x^2:⟹

4

x

2

+

4

x

2

+

9

x

2

+7x+56=x

2

:\implies\sf \dfrac{11 \: x^2}{18} + 7x + 56 = x^2:⟹

18

11x

2

+7x+56=x

2

:\implies\sf 7x + 56 = \dfrac{7 x^2}{18}:⟹7x+56=

18

7x

2

:\implies\sf 7x^2 - 136x - 1008 = 0:⟹7x

2

−136x−1008=0

:\implies\sf x^2 - 18x - 144 = 0:⟹x

2

−18x−144=0

:\implies\sf x^2 - 24x + 6x - 144 = 0:⟹x

2

−24x+6x−144=0

:\implies\sf x(x - 24) + 6(x - 24) = 0:⟹x(x−24)+6(x−24)=0

Now, Comparing these factors with 0.

:\implies\sf x - 24 = 0:⟹x−24=0

:\implies\sf\red{x = 24}:⟹x=24

:\implies\sf x + 6 = 0:⟹x+6=0

:\implies\sf\red{x = -6}:⟹x=−6

__________________

:\implies\sf x^2 \:\:\:\:\:\:\:\:\:\:\:\:\:\:\ [ x = 24]:⟹x

2

[x=24]

:\implies\sf 24 \times 24:⟹24×24

:\implies\sf\pink{576}:⟹576

\mathfrak{\huge{\blue{\underline{\underline{Hemce :}}}}}

Hemce:

Total Number of Birds are = 576.