How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
Learn More

Diamond Pattern | AMC-10A, 2009 | Problem 15

Try this beautiful problem from AMC 10A, 2009 based on Diamond Pattern.

Diamond Pattern - AMC-10A, 2009- Problem 15

The figures \(F_1\), \(F_2\), \(F_3\), and \(F_4\) shown are the first in a sequence of figures. For \(n\ge3\), \(F_n\) is constructed from \(F_{n - 1}\) by surrounding it with a square and placing one more diamond on each side of the new square than \(F_{n - 1}\) had on each side of its outside square. For example, figure \(F_3\) has \(13\) diamonds. How many diamonds are there in figure \(F_{20}\)?

Diamond Pattern
  • \(756\)
  • \(761\)
  • \(786\)

Key Concepts




Check the Answer

Answer: \(761\)

AMC-10A (2009) Problem 15

Pre College Mathematics

Try with Hints

Diamond Pattern

From the above diagram we observe that in \(F_1\) the number of diamond is \(1\).in \(F_2\) the number of diamonds are \(5\).in \(F_3\) the number of diamonds are \(13\). in \(F_4\) the numbers of diamonds are \(25\).Therefore from \(F_1\) to \(F_2\) ,\((5-1)\)=\(4\) new diamonds added.from \(F_2\) to \(F_3\),\((13-5=8\) new diamonds added.from \(F_3\) to \(F_4\),\((25-13)=12\) new diamonds added.we may say that When constructing \(F_n\) from \(F_{n-1}\), we add \(4(n-1)\) new diamonds.

Can you now finish the problem ..........

so we may construct that Let \(S_n\) be the number of diamonds in \(F_n\). We already know that \(P_1\)=1 and for all \(n >1\) ,\(P_n=P_{n-1}+4(n-1)\).now can you find out \(P_{20}\)

can you finish the problem........

Now \(P_{20}\)=\(P_{19} + 4(20-1)\)

\(\Rightarrow P_{20}\)=\(P_{19} + 4.19)\)

\(\Rightarrow P_{20}\)=\(P_{18}+(4 \times 18) +( 4 \times 19)\)

\(\Rightarrow P_{20}\)= .........................................

\(\Rightarrow P_{20}\)=\(1+4(1+2+3+........+18+19)\)

\(\Rightarrow P_{20}\)=\(1+ \frac{ 4 \times 19 \times 20}{2}\)

\(\Rightarrow P_{20}\)=\(761\)

Subscribe to Cheenta at Youtube

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.