Too many Twos

Number Theory Level pending

The addition shown below representing 2 + 22 + 222 + 2222 + ⋯ 2+22+222+2222+\cdots has 101 rows and the last term consists of 101 number of 2's.

What is A + B + C ? A+B+C?

Problem and image courtesy CEMC.
8 6 10 4 2

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

Focussing only on three rightmost columns, their sum will be

2 + 22 + 99 × 222 = 2 + 22 + ( 100 − 1 ) × 222 = 2 + 22 + 99 \times 222 = 2 + 22 + (100 - 1) \times 222 =

2 + 22 + 22200 − 222 = 2 + 22200 − 200 = 22002 2 + 22 + 22200 - 222 = 2 + 22200 - 200 = 22002 .

Thus A + B + C = 2 + 0 + 0 = 2 A + B + C = 2 + 0 + 0 = \boxed{2} .

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Nice solution.

Hana Wehbi - 4 years, 3 months ago

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