Nice Maths Question...

If lucky no. is defined as the no. whose sum of digits is 7, then lucky nos. are in sequence: 7,16,25,34... If 7=A1, 16=A2, 25=A3 and so on; find A65 and A325. (After adding digits once, you cannot add them more times for example:583 is not a lucky no. as you have to add its digits twice to get 7)

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

Stars and bars is applicable here.

For single digit: 7

(1 of them)

For two digits: we have $\overline{a_1 a_2}$ , where $a_1 + a_2=7$ , $0 < a_1 \leq 7$ , $0 \leq a_2 < 7$ which gives $16,25,34,43,52,61,70$ .

(7 of them)

For three digits: we have $\overline{a_1 a_2 a_3}$ , where $a_1 + a_2 + a_3 =7$ , $0 < a_1 \leq 7$ , $0 \leq a_2, a_3 < 7$ .

When $a_1 = 1$ , there's ${ 6+1 \choose 1 }$ solution

When $a_2 = 2$ , there's ${ 5+1 \choose 1 }$ solution

...

when $a_1 = 7$ , there's ${ 0+1 \choose 1 }$ solution.

(28 of them)

For four digits: we have $\overline{a_1 a_2 a_3 a_4}$ , where $a_1 + a_2 + a_3 +a _4=7$ , $0 < a_1 \leq 7$ , $0 \leq a_2, a_3, a_4 < 7$ .

When $a_1 = 1$ , there's ${ 6+2 \choose 2 }$ solution

When $a_1 = 2$ , there's ${ 5+2 \choose 2 }$ solution

...

When $a_1 = 7$ , there's ${ 0+2 \choose 2 }$ solution

(84 of them)

For five digits: we have $\overline{a_1 a_2 a_3 a_4 a_5}$ , where $a_1 + a_2 + a_3 +a _4 +a_5 =7$ , $0 < a_1 \leq 7$ , $0 \leq a_2, a_3, a_4, a_5 < 7$ .

When $a_1 = 1$ , there's ${ 6+3 \choose 3 }$ solution

When $a_1 = 2$ , there's ${ 5+3 \choose 3 }$ solution

...

When $a_1 = 7$ , there's ${ 0+3 \choose 3 }$ solution

(210 of them)

Since $1+7 + 28 < 65 < 1+7 + 28 + 84$ ,

$A_{65}$ must be a four digit number. $65-1-7-28 = 29$ .

$29 - { 6+2 \choose 2 } = 1$ . So $A_{65}$ must the smallest four digit number with $a_1 = 2$ .

$\Rightarrow \boxed{ A_{65} = 2005 }$

Similarly $1+7 + 28 + 84 < 325 < 1+7 + 28 + 84 + 210$ ,

$A_{325}$ must be a five digit number. $325-1-7-28-84 = 205$ . Or $325-1-7-28-84 -210 = -5$

$\Rightarrow A_{330} = 70000$

$\Rightarrow A_{329} = 61000$

$\Rightarrow A_{328} = 60100$

$\Rightarrow A_{327} = 60010$

$\Rightarrow A_{326} = 60001$

$\Rightarrow \boxed{ A_{325} = 52000 }$

Pi Han Goh - 7 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$