Obtain the Generating function for the sequence 4,4,4,4,4,4,4
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Answered by
3
Answer:
S=1+16x+81x
2
+256x
3
+625x
4
+1296x
5
…
−5xS=−5x−80x
2
−405x
3
−1280x
4
−3125x
5
−…
10x
2
S=10x
2
+160x
3
+810x
4
+2560x
5
+…
−10x
3
S=−10x
3
−160x
4
−810x
5
−…
5x
4
S=5x
4
+80x
5
+…
−x
5
S=−x
5
−…
Adding the above series, we get
(1−3x+10x
2
−10x
3
+5x
4
−x
5
)S=1+11x+11x
2
+x
3
⟹(1−x)
5
S=1+11x+11x
2
+x
3
⟹S=
(1−x)
5
1+11x+11x
2
+x
3
Answered by
0
The Generating function for the sequence 4,4,4,4,4,4,4 is given below,
- A generating function is a mathematical tool used in combinatorics to represent a sequence of numbers.
- It is a formal power series that encodes the sequence, where the coefficients of the power series correspond to the terms of the sequence.
- In the case of the sequence 4, 4, 4, 4, 4, 4, 4, since all the terms of the sequence are the same, the generating function will have the same coefficient for each term.
- This allows us to factor out the common factor of 4 and write the generating function as:
G(x) = 4(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)
- The term 4x^k in the generating function corresponds to the kth term in the sequence, which is 4 for all values of k.
- The exponent k in x^k represents the position of the term in the sequence, starting from 0.
- For example, 4x^2 corresponds to the third term in the sequence, which is 4.
- In general, if we have a sequence with different terms, the generating function will have different coefficients for each term, and the coefficients will depend on the values of the terms.
- The generating function can be used to study the properties of the sequence, such as its sum, product, and recurrence relation.
To learn more samples visit,
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