What-is-the-last-digit-from-the-sum-of-1-2-1-2-2-2-3-2-3-50-2-50-

Question Number 21738 by Joel577 last updated on 02/Oct/17

What is the last digit from the sum of

1 . 2^{1} + 2 . 2^{2} + 3 . 2^{3} + \dots + 50 . 2^{50} ?

Answered by $@ty@m last updated on 02/Oct/17

S_{50} = 1 . 2^{1} + 2 . 2^{2} + 3 . 2^{3} + \dots + 50 . 2^{50}

2 . S_{50} = 1 . 2^{2} + 2 . 2^{3} + \dots \dots + 49 . 2^{50} + 50 . 2^{51}

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S u b t r a c t i n g

- S_{50} = 2^{1} + 2^{2} + 2^{3} + \dots \dots + 2^{50} - 50 . 2^{51}

- S_{50} = \frac{2 (2^{50} - 1)}{2 - 1} - 50 . 2^{51}

- S_{50} = 2^{51} - 2 - 50 . 2^{51}

S_{50} = 49 . 2^{51} + 2 - - - (1)

w e o b s e r v e t h a t

l a s t d i g i t i n 2^{1} = 2

l a s t d i g i t i n 2^{2} = 4

l a s t d i g i t i n 2^{3} = 8

l a s t d i g i t i n 2^{4} = 6

l a s t d i g i t i n 2^{5} = 2

.

.

\dots a n d s o o n .

\Rightarrow l a s t d i g i t i n 2^{4 n + 3} = 8

\Rightarrow l a s t d i g i t i n 2^{4 \times 12 + 3} = 8

\Rightarrow l a s t d i g i t i n 49 . 2^{51} + 2 = 4 A n s .

Commented by Joel577 last updated on 03/Oct/17

T h a n k s . B u t m y f r i e n d g o t 8

H e u s e d m o d t o f i n d t h e l a s t d i g i t

Commented by $@ty@m last updated on 03/Oct/17

I n t h a t c a s e e i t h e r h e o r I a m w r o n g .

T r y t o f i n d w h e r e i s t h e m i s t a k e .

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