Determine-nth-term-of-the-following-sequence-1-0-1-0-7-28-79-

Question Number 3061 by Rasheed Soomro last updated on 04/Dec/15

Determine nth term of the following sequence

1, 0, - 1, 0, 7, 28, 79, \dots

Commented by prakash jain last updated on 04/Dec/15

o n e t e c h i n q u e i s t o d e f i n e

d e f i n e a s e r i e s b y s u c c e s s i v e s u b t r a c t i o n

u n t i l y o u g e t a k n o w n p a t t e r n .

a_{1} = 1

a_{2} = 0

a_{3} = - 1

a_{4} = 0

a_{5} = 7

a_{6} = 28

a_{7} = 79

b_{2} = a_{2} - a_{1} = - 1

b_{3} = - 1

b_{4} = 1, b_{5} = 7, b_{6} = 21, b_{7} = 51

s i m i l a r y c_{n} = b_{n} - b_{n - 1}

c_{3} = 0 c_{4} = 2 c_{5} = 6 c_{6} = 14 c_{7} = 30

d_{n} = c_{n} - c_{n - 1}

d_{4} = 2 d_{5} = 4 d_{6} = 8 d_{7} = 16

d_{n} = 2^{n - 3} (n ⩾ 4)

c_{n} = c_{n - 1} + d_{n} (n ⩾ 4)

b_{n} = b_{n - 1} + c_{n} (n ⩾ 3)

a_{n} = a_{n - 1} + b_{n} (n ⩾ 2)

f o r 8^{t h} t e m

d_{8} = 32

c_{8} = c_{7} + d_{8} = 30 + 32 = 62

b_{8} = b_{7} + c_{8} = 51 + 62 = 113

a_{8} = a_{7} + b_{8} = 79 + 113 = 192

Commented by Rasheed Soomro last updated on 05/Dec/15

TH a n k S^{f o r} \overset{V a l u e a b l e}{G u i d a n c e}!

Answered by Filup last updated on 04/Dec/15

1 = 2 - 1 \Rightarrow 2^{1} - 1

0 = 4 - 4 \Rightarrow 2^{2} - 2^{2}

- 1 = 8 - 9 \Rightarrow 2^{3} - 3^{2}

e t c .

T_{n} = 2^{n} - n^{2}

Commented by Rasheed Soomro last updated on 04/Dec/15

V^{e r y} G^{OO} D!

Commented by RasheedAhmad last updated on 04/Dec/15

H o w d i d y o u r e a c h t h a t s o l u t i o n!

I s t h e r e a w a y ? I t h i n k t h e s e t y p e s

o f q u e s t i o n s l i k e^{'} r i d d l e s^{'} w h i c h

c a n b e s o l v e d o n l y b y g u e s s e s!

A l t h o u g h i n t e l l i g e n t g u e s s e s!

Commented by prakash jain last updated on 04/Dec/15

See comments in question about approach

to find general terms . Uniqueness of

formula is not guaranteed when only finite

number of terms are known .

Also see quesion 3003, where two different

formulas can generate all known terms

correctly but differ on next term .