Question

Determine a formula involving binomial coefficients for the nth term of a sequence if its initial terms are those listed. (Hint: Looking at Pascal’s triangle will be helpful. Although infinitely many sequences start with a specified set of terms, each of the following lists is the start of a sequence of the type desired.)

\(\begin{array}{l}a) 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, . . .\\b) 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, . . .\\c) 1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, . . .\\d) 1, 1, 2, 3, 6, 10, 20, 35, 70, 126, . . .\\e) 1, 1, 1, 3, 1, 5, 15, 35, 1, 9, . . .\\f ) 1, 3, 15, 84, 495, 3003, 18564, 116280, 735471, 4686825, . .\;.\end{array}\)

\(a) 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, . . .\)

Short answer

Therefore, the required binomial coefficient for the nth term of a sequence if its initial terms are those listed is \(\left( {\begin{array}{*{20}{c}}n\\2\end{array}} \right),n \ge 2\).

Step-by-step solution

Step 1: Explanation

Given:

Initial terms:

\({\rm{1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, }}{\rm{. }}{\rm{. }}{\rm{.}}\)

To find:

The objective is to determine a formula involving binomial coefficients for the nth term of a sequence if its initial terms are those listed.

Step 2: Rewrite and further simplification

We can rewrite \({\rm{1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, }}{\rm{. }}{\rm{. }}{\rm{.}}\) as

\(\begin{array}{l}C_0^1,C_2^3,C_2^4,C_2^5,C_2^6,C_2^7,C_2^8,C_2^9,C_2^{10},C_2^{11},........\\ = C_2^2,C_2^3,C_2^4,C_2^5,C_2^6,C_2^7,C_2^8,C_2^9,C_2^{10},C_2^{11},........\;\;\;\;(As\;C_r^n = C_{n - r}^n)\end{array}\)

The nth term of the sequence is \(\left( {\begin{array}{*{20}{c}}n\\2\end{array}} \right),n \ge 2\)

Therefore, the required binomial coefficient for the nth term of a sequence if its initial terms are those listed is \(\left( {\begin{array}{*{20}{c}}n\\2\end{array}} \right),n \ge 2\).

\(b) 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, . . .\)

Answer

Therefore, the required binomial coefficient for the nth term of a sequence if its initial terms are those listed is \(\left( {\begin{array}{*{20}{c}}n\\3\end{array}} \right),n \ge 3\).

Step 3: Explanation

Given:

Initial terms:

\(1, 4, 10, 20, 35, 56, 84, 120, 165, 220, . . .\)

To find:

The objective is to determine a formula involving binomial coefficients for the nth term of a sequence if its initial terms are those listed.

Step 4: Rewrite and further simplification

We can rewrite \(1, 4, 10, 20, 35, 56, 84, 120, 165, 220, . . .\) as

\(C_3^3,C_3^4,C_3^5,C_3^6,C_3^7,C_3^8,C_3^9,C_3^{10},C_3^{11},C_3^{12},........\;\)

The nth term of the sequence is \(C_3^n\)

\(\left( {\begin{array}{*{20}{c}}n\\3\end{array}} \right),n \ge 3\)

Therefore, the required binomial coefficient for the nth term of a sequence if its initial terms are those listed is \(\left( {\begin{array}{*{20}{c}}n\\3\end{array}} \right),n \ge 3\).

\(c) 1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, . . .\)

Answer

Therefore, the required binomial coefficient for the nth term of a sequence if its initial terms are those listed is \(\left( {\begin{array}{*{20}{c}}{2n - 2}\\{n - 1}\end{array}} \right),n \ge 1\).

Step 5: Explanation

Given:

Initial terms:

\(1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, . . .\)

To find:

The objective is to determine a formula involving binomial coefficients for the nth term of a sequence if its initial terms are those listed.

Step 6: Rewrite and further simplification

We can rewrite \(1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, . . .\) as

\(C_0^0,C_1^2,C_2^4,C_3^6,C_4^8,C_5^{10},C_6^{12}........\;\)

The nth term of the sequence is \(C_n^{2n},n \ge 1\)

We do get \(C_0^0\) from \(C_n^{2n},n \ge 1\)

\(\left( {\begin{array}{*{20}{c}}{2n - 2}\\{n - 1}\end{array}} \right),n \ge 1\)

Therefore, the required binomial coefficient for the nth term of a sequence if its initial terms are those listed is \(\left( {\begin{array}{*{20}{c}}{2n - 2}\\{n - 1}\end{array}} \right),n \ge 1\).

\(d) 1, 1, 2, 3, 6, 10, 20, 35, 70, 126, . . .\)

Answer

Therefore, the required binomial coefficient for the nth term of a sequence if its initial terms are those listed is \(\left( {\begin{array}{*{20}{c}}n\\{\frac{n}{2}}\end{array}} \right),n \ge 0\).

Step 7: Explanation

Given:

Initial terms:

\(1, 1, 2, 3, 6, 10, 20, 35, 70, 126, . . .\)

To find:

The objective is to determine a formula involving binomial coefficients for the nth term of a sequence if its initial terms are those listed.

Step 8: Rewrite and further simplification

The binomial coefficient: it is the number of ways of picking unordered outcomes from possibilities, also known as a combination or combinatorial number.

\(\left( {\begin{array}{*{20}{c}}n\\{\frac{n}{2}}\end{array}} \right),n \ge 0\)

Therefore, the required binomial coefficient for the nth term of a sequence if its initial terms are those listed is \(\left( {\begin{array}{*{20}{c}}n\\{\frac{n}{2}}\end{array}} \right),n \ge 0\).

\(e) 1, 1, 1, 3, 1, 5, 15, 35, 1, 9, . . .\)

Answer

Therefore, the required binomial coefficient for the nth term of a sequence if its initial terms are those listed is \(\left( {\begin{array}{*{20}{c}}n\\{n(mode\;4)}\end{array}} \right),n \ge 0\).

Step 9: Explanation

Given:

Initial terms:

\(1, 1, 1, 3, 1, 5, 15, 35, 1, 9, . . .\)

To find:

The objective is to determine a formula involving binomial coefficients for the nth term of a sequence if its initial terms are those listed.

Step 10: Rewrite and further simplification

The binomial coefficient: it is the number of ways of picking unordered outcomes from possibilities, also known as a combination or combinatorial number.

\(\left( {\begin{array}{*{20}{c}}n\\{n(mode\;4)}\end{array}} \right),n \ge 0\)

Correction: 4^th term in the question should be one instead of three

Therefore, the required binomial coefficient for the nth term of a sequence if its initial terms are those listed is \(\left(

{\begin{array}{*{20}{c}}n\\{n(mode\;4)}\end{array}} \right),n \ge 0\).

\(f ) 1, 3, 15, 84, 495, 3003, 18564, 116280, 735471, 4686825, . .\;.\)

Answer

Therefore, the required binomial coefficient for the nth term of a sequence if its initial terms are those listed is \(\left( {\begin{array}{*{20}{c}}{3n}\\n\end{array}} \right),n \ge 0\).

Step 11: Explanation

Given:

Initial terms:

\(1, 3, 15, 84, 495, 3003, 18564, 116280, 735471, 4686825, . .\;.\)

To find:

The objective is to determine a formula involving binomial coefficients for the nth term of a sequence if its initial terms are those listed.

Step 2: Rewrite and further simplification

The binomial coefficient: it is the number of ways of picking unordered outcomes from possibilities, also known as a combination or combinatorial number.

\(\left( {\begin{array}{*{20}{c}}{3n}\\n\end{array}} \right),n \ge 0\)

Correction: 4^th term in the question should be one instead of three

Therefore, the required binomial coefficient for the nth term of a sequence if its initial terms are those listed is \(\left( {\begin{array}{*{20}{c}}{3n}\\n\end{array}} \right),n \ge 0\).