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“There Are No Limits To What You Can Accomplish, Except The Limits You Place On Your Own Thinking.” – Brian Tracy

Meaning of Sequences, Meaning of General Term (nth term), Meaning of Series

NCERT Exercise 9.1
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Write the first five terms of each of the sequences in Exercises 1 to 6 whose nth terms are:

Question 1. \( a_n = n(n+2) \)
Question 2. \( a_n = \frac{n}{n+1}\)
Question 3. \( a_n=2^n\)
Question 4. \( a_n=\frac{2n-3}{6}\)
Question 5. \( a_n=(-1)^{n-1} 5^{n+1}\)
Question 6. \( a_n=n \frac{n^2+5}{4}\)

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Find the indicated terms in each of the sequences in Exercises 7 to 10 whose nth terms are:

Question 7. \( a_n=4n-3; a_{17}, a_{24}\)
Question 8. \( a_n=\frac{n^2}{a^n}; a_7\)
Question 9. \( a_n=(-1)^{n-1}n^3; a_9\)
Question 10. \( a_n=\frac{n(n-2)}{n+3}; a_{20}\)

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Write the first five terms of each of the sequences in Exercises 11 to 13 and obtain the corresponding series:

Question 11. \( a_1=3, a_n=3a_{n-1}+2\) for all n>1
Question 12. \( a_1=-1, a_n=\frac{a_{n-1}}{n}, n \ge 2\)
Question 13. \( a_1=a_2=2, a_n=a_{n-1}-1, n>2\)
Question 14. The Fibonacci sequence is defined by
\( 1=a_1=a_2 \text{ and } a_n=a_{n-1}+a_{n-2}, n>2\)
Find \( \frac{a_{n+1}}{a_n}\) for n=1, 2, 3, 4, 5.

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