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I was doing some proof by induction and I stumbled upon this general statement that I can't derive to save my life:

$$\frac{k(k+1)}{2} + (k+1) = \frac{(k+1)(k+2)}{2}$$

Could you prove why it is true (don't verify it please I've already done that)

Alex Jones
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Yusuf Abukar
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  • Use MathJaX. Your expression is hard to read and then almost impossible to parse: what's in the numerator, what in the denominator...? More than one month a member, you should by now try to write properly mathematics in this site. – DonAntonio Apr 18 '17 at 08:33
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    @DonAntonio write proper mathematics (sorry I had to do it) – DHMO Apr 18 '17 at 08:36
  • @DonAntonio I don't know how to do it on mobile – Yusuf Abukar Apr 18 '17 at 12:49
  • @DHMO English is no my mother tongue and, in fact, not even the second on but the third one, but I think you can write "do that properly" or "write properly", and thus "write properly mathematics", or perhaps "write mathematics properly", as opposed to "proper mathematics", which has nothing to do with writing or not. – DonAntonio Apr 18 '17 at 12:58

2 Answers2

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$$\dfrac{k(k+1)}2+k+1=\dfrac{k(k+1)}2+2\cdot\dfrac{(k+1)}2=\dfrac{k+1}2\cdot(k+2)$$

lab bhattacharjee
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\begin{align}\frac{k(k+1)}2+(k+1)&=\frac{k(k+1)}2+\frac{2(k+1)}2\tag{1}\\ &=\frac{k(k+1)+2(k+1)}2\tag{2}\\ &= \frac{(k+1)(k+2)}2\tag{3}\end{align}

Justifications:

$(1)$ - common denominator of $2$

$(2)$ - combine fractions

$(3)$ - factor out $(k+1)$

lioness99a
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