a-1-a-2-a-3-a-n-is-a-sequence-satifies-that-a-n-2-a-n-1-a-n-for-n-1-suppose-the-sum-of-the-first-999-terms-1003-and-the-sum-of-the-first-1003-terms-999-find-the-

Question Number 192597 by York12 last updated on 22/May/23

a_{1}, a_{2}, a_{3}, \dots ., a_{n} i s a s e q u e n c e s a t i f i e s t h a t

a_{n + 2} = a_{n + 1} - a_{n} f o r n \geq 1 . s u p p o s e t h e s u m

o f t h e f i r s t 999 t e r m s = 1003 a n d t h e s u m

o f t h e f i r s t 1003 t e r m s = - 999 f i n d t h e

s u m o f t h e f i r s t 2002 t e r m s .

Answered by MM42 last updated on 22/May/23

a_{4} = a_{3} - a_{2} = - a_{1} & a_{5} = a_{4} - a_{3} = - a_{2} & a_{6} = a_{5} - a_{4} = - a_{3}

\Rightarrow a_{6 k + 1} = a_{1} & a_{6 k + 2} = a_{2} & a_{6 k + 3} = a_{3}

a_{6 k + 4} = - a_{1} & a_{6 k + 5} = - a_{2} & a_{6 k} = - a_{3}

999 = 6 \times 166 + 3 \Rightarrow S_{999} = a_{1} + a_{2} + a_{3} = 2 a_{2} = 1003 \Rightarrow a_{2} = 501.5

1003 = 6 \times 167 + 1 \Rightarrow S_{1003} = a_{1} \Rightarrow a_{1} = - 999

\Rightarrow a_{3} = 1500.5

2002 = 6 \times 336 + 4 \Rightarrow S_{2002} = a_{1} + a_{2} + a_{3} + a_{4} = a_{2} + a_{3} = 2002

Commented by York12 last updated on 22/May/23

a_{2} = 501.5

s_{2002} = 999 + 2 \times 501.5 = 2002

Commented by York12 last updated on 22/May/23

a_{4} = a_{3} - a_{2} = - a_{1} & a_{5} = a_{4} - a_{3} = - a_{2} & a_{6} = a_{5} - a_{4} = - a_{3}

\Rightarrow a_{6 k + 1} = a_{1} & a_{6 k + 2} = a_{2} & a_{6 k + 3} = a_{3}

a_{6 k + 4} = - a_{1} & a_{6 k + 5} = - a_{2} & a_{6 k} = - a_{3}

999 = 6 \times 166 + 3 \Rightarrow S_{999} = a_{1} + a_{2} + a_{3} = 2 a_{2} = 1003 \Rightarrow a_{2} = 501.5

1003 = 6 \times 167 + 1 \Rightarrow S_{1003} = a_{1} \Rightarrow a_{1} = - 999

⇛ a_{3} = 1500.5

2002 = 6 \times 336 + 4 \Rightarrow S_{2002} = a_{1} + a_{2} + a_{3} + a_{4} = a_{2} + a_{3} = 2002

Facebook Tweet Pin