Question

Let ${a_n} $be a sequence given by: $\{ 1,6,15,28,45,66,......,f(n)\} $. Show that the generating function $f(n) $is of the form $a{n^2} + bn + c. $Find the formula by computing the coefficients a, b, c?

Answer 1

Hint: First of all we will find the sequence pattern and then follow the pattern to find the unknowns in the given form of the equation. Follow the step by step approach to get the pattern of the given sequence.

Complete step-by-step answer:
Let us take the given sequence:
 ${P_n} = \{ 1,6,15,28,45,66,......,f(n)\} $
Now find the difference between two consecutive terms and write the resultant values.
 $\{ 1,5,9,13,17,21,......,\} $
Now again find the difference between the two consecutive terms.
 $\{ 4,4,4,4,4,......,\} $
In discrete mathematics to find the difference is the same as taking the derivative that is slope. We had used the difference application twice before reaching the constant number $4 $which suggests that the sequence is of the polynomial growth.
Given that assert that ${P_n} = a{n^2} + bn + c. $
Now, all we have to do is to find the values for a, b and c.
Now to get the values for a, b and c, first of all we will use the three entry of the sequence setting as $n = \{ 1,2,3\} $
For Example:
 Equation $1 \Rightarrow {P_1} = a + b + c = 1 $
Similarly, equation $2 \Rightarrow {P_2} = 4a + 2b + c = 6 $
And equation $3 \Rightarrow {P_3} = 9a + 3b + c = 15 $
Multiply equation $1 $ with the number $4 $
 $ \Rightarrow 4a + 4b + 4c = 4 $
Subtract equation $2 $from the above equation.
 $ \Rightarrow 2b + 3c = - 2 $ …. (A)
Now, again multiply equation $1 $ with the number $9 $
 $ \Rightarrow 9a + 9b + 9c = 9 $
Subtract equation $3 $from the above equation.
 $ \Rightarrow 6b + 8c = - 6 $ …. (B)
Now, multiply equation (A) with the number $3 $
 $ \Rightarrow 6b + 9c = - 6 $
Subtract equation (B) from the above equation:
 $ \Rightarrow - c = ( - 6) - ( - 6) $
Remember minus minus is plus.
 $ \Rightarrow - c = - 6 + 6 $
Remember when you are simplifying one positive number and one negative number, you have to do subtraction, but sign of a bigger number.
 $ \Rightarrow - c = 0 $
Multiply minus one on both the sides of the equation.
 $ \Rightarrow c = 0 $ …. (C)
Place the value of c in the equation (B)
 $ \Rightarrow 6b + 8(0) = - 6 $
Zero multiplied with any number gives zero as the resultant value. Simplify the above equation-
 $ \Rightarrow 6b + 0 = - 6 $
Term multiplicative on one side, if moved to opposite then it goes to the denominator.
 $ \Rightarrow b = - \dfrac{6}{6} $
 $ \Rightarrow b = - 1 $
Now, place the values of “b” and “c” in the equation $1 $
 $a + b + c = 1 $
By placing the values-
 $ \Rightarrow a - 1 + 0 = 1 $
Take all the constants on the right hand side of the equation
 $ \Rightarrow a = 1 + 1 $
Simplification
 $ \Rightarrow a = 2 $
Therefore, $\left[ {\begin{array}{*{20}{c}}
  a \\
  b \\
  c
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  2 \\
  { - 1} \\
  0
\end{array}} \right] $
Hence, we can rewrite in terms of n as: ${P_n}^6 = 2{n^2} - n+ 0 $
Verify placing the values of $n = 1,2,3 $
Which implies
  $
  {P_1}^6 = 1 \\
  {P_2}^6 = 6 \\
  {P_3}^6 = 15 \;
  $
This is the required solution.
So, the correct answer is “ ${P_n}^6 = 2{n^2} - n+ 0 $”.

Note: You can find the values of the unknowns by using the matrix method, here we have used a combination of elimination and substitution method. Be careful about the sign convention. Always remember that when you multiply any term with any number it should be multiplied with all the terms on both the sides of the equation for its equivalent value.