f(1)=1, f(n)=2Σ_(k=1) ^(n−1) f(k). find Σ


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Question Number 183064 by Matica last updated on 19/Dec/22

$f (1) = 1, f (n) = 2 \underset{k = 1}{\overset{n - 1}{Σ}} f (k) . f i n d \underset{k = 1}{\overset{m}{Σ}} f (k) .$
	Answered by mr W last updated on 19/Dec/22

	$a_{n} = f (n)$ $s_{n} = \underset{k = 1}{\sum^{n}} a_{k}$ $a_{n} = 2 (a_{1} + a_{2} + . . . + a_{n - 1})$ $3 a_{n} = 2 (a_{1} + a_{2} + . . . + a_{n - 1} + a_{n}) = 2 s_{n}$ $a_{n} = \frac{2}{3} s_{n}$ $s_{n} - s_{n - 1} = \frac{2}{3} s_{n}$ $s_{n} = 3 s_{n - 1} = 3^{2} s_{n - 2} = 3^{n - 1} s_{1} = 3^{n - 1} a_{1} = 3^{n - 1}$ $\underset{k = 1}{\sum^{m}} f (k) = s_{m} = 3^{m - 1} ✓$
	Commented by Matica last updated on 19/Dec/22

	$T h a n k a l o t .$
	Commented by Matica last updated on 19/Dec/22

	$H o w d i d y o u g e t s_{1} = a_{1} ? w h y n o t a_{1} = \frac{2}{3} s_{1} ?$
	Commented by mr W last updated on 19/Dec/22

	$s_{n} m e a n s t h e s u m o f f i r s t n t e r m s .$ $a_{1} = 1, s_{1} = a_{1} = 1$ $t h e s e a r e g i v e n i n i t i a l c o n d i t i o n s .$ $s_{1} i s t h e s u m o f t h e f i r s t t e r m, i t m u s t$ $b e a_{1}! o r d o y o u t h i n k a_{1} = \frac{2}{3} s_{1} m a k e s$ $a n y s e n s e ?$
	Commented by Matica last updated on 19/Dec/22

	$O k, s i r . i d e c i d e d .$
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