If (1 + x + x 2 ) n = a 0 + a 1 x + a 2 x 2 + ......+ a 2n x n , then the value of a 0 + a 3 + a 6 +.........is -
Text Solution
Verified by Experts(iv) (i); (v) (iii); (vi) (v)
(a, b, c)
(1 + x + x 2 ) n = a 0 + a 1 x + a 2 x 2 +.......+a 2n x 2n
Now x = w
(1 + w + w 2 ) n = a 0 +a 1 w + a 2 w 2 +......+a 2n w 2n .... (i)
x = w 2 (1+w+w
2 ) n = a 0 + a 1 w 2 + a 2 w 4 +.....+a 2n w 4n .....(ii)
x = 1
3 n = a 0 + a 1 + a 2 + ........+a 2n ..............(iii)
adding (i), (ii) & (iii)
3 n = 3(a 0 + a 3 + a 6 + ..................)
⇒ a 0 + a 3 + a 6 +..................=3 n–1 .............(iv)
(i) – (ii)
⇒ (w – w 2 ) (a 1 – a 2 + a 4 – a 5 +...) = 0.......(v)
(iii) – (iv)
⇒ a 1 + a 2 + a 4 + a 5 +..........= 3 n –3 n–1 = 2 . 3 n–1 ........(vi)
(v) + (vi)
⇒ 2(a 1 + a 4 + a 7 +...........) = 2 . 3 n –1
(vi) – (v)
⇒ 2(a 2 + a 5 + ..........) = 2.3 n–1
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