Sequences and Series

a², b², c² . A.P.
a² + 1, b² + 1, c² + 1 . P.
a² + ab + bc + ca, b² + ab + bc + ca, c² + ab + bc + CA . P.
⇒ (a + b) (a + c), (a + b) (b + c), (b + c) (c + a) . P.
⇒ 1/ (b+c), 1/ (c+a), 1/ (a+b) . A.P.
⇒ b + c, c + a, a + b . P.

First element in 20^th group is equal to '20'.

(20, 21, ….upto 39 times)

$\therefore Sum=\frac{39}{2}\left(2*20+\left(39-1\right)*1\right)$        

log? (abc) = 6 ⇒ abc = 6?
a = b/r & c = br
⇒ b = 36 and a = 36/r
⇒ r = 2,3,4,6,9,12,18
Also b - a = 36 (1 - 1/r) is a perfect cube.
∴ r = 4 ⇒ a + b + c = 9 + 36 + 144 = 189

  $L=\underset{x\to \infty }{lim}\frac{{\left(2+\frac{1}{x}\right)}^{40}{\left(4-\frac{1}{x}\right)}^5}{{\left(2+\frac{3}{x}\right)}^{45}}=32$

lim (x→0) (ae²? - bcosx + c) / (x sinx)
= lim (x→0) (ae²? - bcosx + c) / x² * lim (x→0) x/sinx
For limit to exist
a - b + c = 0
2a + b = 0
(4a-b)/2 = 1
c = 2, b = 2, a = -1
a+b+c = 3

T? = (1/3)T?
⇒ T? , T? , T? G.P.
T = 1/3
T? = 1/243
ar? = 1/243
a (1/3)? = 1/243
a/729 = 1/243 ⇒ a = 3

Sum of infinite series = T? + T? + T?
= (T? ) / (1-r²) = (3 * 1) / (1 - 1/9) = 3 / (8/9) = 27/8

b? , b? , b?
a, ar, ar².
a + ar = 1 ⇒ a = 1 / (1+r)

Σ (from k=1 to ∞) b? = 2 ⇒ a / (1-r) = 2
(1 / (1+r) / (1-r) = 2
1 / (1-r²) = 2
1/2 = 1 - r²
r² = 1/2 ⇒ r = ±1/√2

b? < 0 r = -1/2
⇒ a = b? = 2 + √2

$t_n=\frac{3^n-2^n}{3^n}=1-{\left(\frac{2}{3}\right)}^n$

$S_{100}=100-\frac{\frac{2}{3}\left({\left(\frac{2}{3}\right)}^{100}-1\right)}{\frac{2}{3}-1}=100+2.\frac{\left(2^{100}-3^{100}\right)}{3^{100}}$

$=100-2+\frac{2^{101}}{3^{100}}=98+{\left(\frac{2}{3}\right)}^{100}.2<98+1$

$e^{\left(co{s}^{2}x+co{s}^{4}x+co{s}^{6}x+.....\infty \right)}lo{g}_{e}2$  

$=e^{\frac{co{s}^{2}x}{1-co{s}^{2}x}lo{g}_{e}2}=e^{co{t}^{2}xlo{g}_{e}2}=2^{co{t}^{2}x}$          

t² – 9t + 8 = 0

(t – 8) (t – 1) = 0

$t=2^{co{t}^{2}x}=8=2^3$

$\Rightarrow co{t}^{2}x=3=co{t}^2\frac{\pi }{6}$

$\frac{2sinx}{sinx+\sqrt[]{3}cosx}=\frac{2*\frac{1}{2}}{\frac{1}{2}+\sqrt[]{3}*\frac{\sqrt[]{3}}{2}}=\frac{1}{2}$            

T? = r/ (2r²)²+1)
= r/ (2r²+1)²- (2r)²)
= (1/4) * (4r)/ (2r²+2r+1) (2r²-2r+1)
S? = 1/4 Σ? ¹? [ 1/ (2r²-2r+1) - 1/ (2r²+2r+1) ]
=> S? = (1/4) * (220/221) = 55/221 = m/n
∴ m+n = 276