MO Z6 I-3 2017 jars

Jano had 100 identical preserving jars, from which he built triangular pyramids. The highest floor of the pyramid always has one jar, the second floor from the top represents an equilateral triangle, whose side consists of two jars, etc. An example of the construction of a three-floor pyramid is in the picture.


1. How many jars did Jano need for a five-floor pyramid?
2. How many floors did the pyramid have on which as many of Jano's jars as possible were used?

Final Answer:

a =  35
b =  7

Step-by-step explanation:

$$\begin{gathered} a_1=1 \\ {a}_2=2+1=3 \\ {a}_3=3+2+1=6 \\ {a}_4=4+3+2+1=10 \\ {a}_5=5+4+3+2+1=15 \\ a=a_1+a_2+a_3+a_4+a_5=1+3+6+10+15=35 \end{gathered}$$

$$\begin{gathered} a_6=6+a_5=6+15=21 \\ {s}_6=a+a_6=35+21=56 \\ {a}_7=7+a_6=7+21=28 \\ {s}_7=s_6+a_7=56+28=84 \\ {a}_8=8+a_7=8+28=36 \\ {s}_8=s_7+a_8=84+36=120 \\ {s}_7<100 \\ b=7 \end{gathered}$$

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