Chapter MPS, Problem 11.11MPS

To determine

Make a conjecture about the sums of the rows.

Answer to Problem 11.11MPS

  $2^n$

Explanation of Solution

Given information:

Study the first eight rows of Pascal’s triangle.

Calculation:

Write the Pascal’s triangle;

  $\begin{array}{cccccccccccccccccc}Row0& & & & & & & & & 1& & & & & & & & \\ Row1& & & & & & & & 1& & 1& & & & & & & \\ Row2& & & & & & & 1& & 2& & 1& & & & & & \\ Row3& & & & & & 1& & 3& & 3& & 1& & & & & \\ Row4& & & & & 1& & 4& & 6& & 4& & 1& & & & \\ Row5& & & & 1& & 5& & 10& & 10& & 5& & 1& & & \\ Row6& & & 1& & 6& & 15& & 20& & 15& & 6& & 1& & \\ Row7& & 1& & 7& & 21& & 35& & 35& & 21& & 7& & 1& \end{array}$

Now find the sum of the terms for each row:

  $\begin{array}{l}Row0:1\\ R0w1:1+1=2\\ Row2:1+2+1=4\end{array}$

  $\begin{array}{l}Row3:1+3+3+1=8\\ Row4:1+4+6+4+1=16\\ Row5:1+5+10+10+5+1=32\\ Row6:1+6+15+20+15+6+1=64\\ Row7:1+7+21+35+35+21+7+1=128\end{array}$

Write the sums of the terms in the first eight rows as shown below:

  $\begin{array}{l}Row0:1=2^0\\ Row1:2=2^1\\ Row2:4=2^2\\ Row3:8=2^3\\ Row4:16=2^4\\ Row5:32=2^5\\ Row6:64=2^6\\ Row7:128=2^7\end{array}$

Hence, if we have $n$ rows then the sum of terms in the $n^{th}$ row is $2^n$ .