# When is the sum of two series equal

Anne and Hestor are both reading an 1100 page historical novel. Anne reads 50 pages a day. Hestor reads 10 pages the first day , 20 the second, 30 the third and so on. After how many days will they be on the same page. When I solve this problem I get 9 days. The books says the answer is 14 days. Which answer is correct?

$$50 \times 9 = 5(9)^2 + 5(9) = 450$$

$10+20+30+40+50+60+70+80+90=450=9\times 50$, so you are correct.

• Downvoters should note that these sort of answers are also welcome on this site. Sometimes these sort of answers somehow look more beautiful. Mar 12, 2016 at 17:45
• What about $$10+20+30+40+50+60+70+80+90\\=\\50+50+50+50+50+50+50+50+50$$ ?
– user65203
Mar 12, 2016 at 17:47
• @YvesDaoust Everything in moderation, including moderation. Mar 12, 2016 at 17:49

You are correct. The pages that Anne reads after k days is given by: $50k$.

The pages that Hector reads after k days is given by: $10\cdot \sum\limits_{n=1}^{k} n$ = $10 \cdot \frac{k(k+1)}{2}$ = $5k(k+1)$.

If you set these two equal, you will get $10 = k+1$, and therefore k = 9, meaning after 9 days they will have read the same amount.

Arithmetic progression for Hestor:

$$a_1=10\;,\;\;d=10\implies a_n=10+(n-1)10=10n$$

Thus the question is when

$$\overbrace{50n}^{\text{No. of pages Anne read}}=\overbrace{S_n}^{\text{No. of pages Hestor read}}=\frac n2(2a_1+(n-1)d)\implies 100=20+(n-1)10\implies$$

$$n-1=8\implies n=9$$

You are correct, the book is wrong...or you read the wrong question's answer.