MCQ
$(444 ..... n $ વખત $ 4)$ + $(666 ..... n $ વખત $ 6)^2$ $=$……………
- A$\frac{4}{9}\left( {{10}^{n}}-1 \right)$
- B$\frac{4}{9}{{\left( {{10}^{n}}-1 \right)}^{2}}$
- C$\frac{9}{4}.10^n( {10^n-1^2} )$
- ✓$\frac{4}{9}.10^n( {10^n}-1 )$
$(444 ..... n $ વખત $ 4) = (4 + 40 +400 + 4000 +.... +400+... n )$ વખત
$= \frac{4 (10^n - 1)}{9}$
$(666 ..... n $ વખત $6) = (6 + 60 +600 + 6000 +....+ 600 + ..... + n$ વખત $) $
$= \frac{6 (10^n - 1)}{9} = \frac{2}{3} (10^n - 1)$
$(666 ...... n $ વખત $6)^2 = \frac{4}{9} (10^n - 1)^2$
$\therefore (444..... n $ વખત $ 4) + (666 ...... n $ વખત $6)^2$
$= \frac{4}{9} (10^n - 1) + \frac{4}{9} (10^n - 1)^2$
$= \frac{4}{9} (10^n - 1) (1 + 10^n - 1) $
$= \frac{4}{9} 10^n (10^n - 1)$
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