# shortcut???????

• Jan 4th 2013, 11:47 AM
vjdm
shortcut???????
what would be the no of digits in 461+462+463+464+465
• Jan 4th 2013, 12:29 PM
HallsofIvy
Re: shortcut???????
All of those numbers is less than or equal to $4^{65}$ so their sum is less than $5(4^{65})$. $65 log(4)= 39.1339$ so $4^{65}$ has 39 digits and 5 times that has 40 digits.
• Jan 4th 2013, 07:09 PM
Soroban
Re: shortcut???????
Hello, vjdm!

Quote:

$\text{What is the number of digits in: }\:N \;=\;4^{61}+4^{62}+4^{63}+4^{64}+4^{65}$

We have: . $N \;=\;4^{61}(1 + 4 + 4^2+4^3+4^4) \;=\;4^{61}\cdot341$

Take logs: . $\log N \;=\;\log\left(4^{61}\cdot341\right) \;=\;\log\left(4^{61}\right) + \log(341)$

m . . . . . . . $\log N \;=\;61\log(4) + \log(341) \;=\;39.25841385$

. . . . . . . . . . . $N \;=\;1.813066986 \times 10^{39}$

Therefore, $N$ has 40 digits.