Sequences and Series
Numbers ki ek nizam (pattern) wali list ko sequence kehte hain, aur unhe jodne par jo banta hai use series. Is chapter me do important patterns padhenge — Arithmetic Progression (AP) aur Geometric Progression (GP) — aur unke sums.
1. Sequence aur Series
- Sequence: ek order me likhe numbers — \(a_1, a_2, a_3, \dots\) Yahan \(a_n\) = \(n\)-th term.
- Series: sequence ke terms ka sum — \(a_1 + a_2 + a_3 + \dots\)
2. Arithmetic Progression (AP)
AP me har consecutive terms ka antar (difference) constant hota hai, jise common difference \(d\) kehte hain. First term \(a\).
AP aisi dikhti hai: \(a,\ a+d,\ a+2d,\ a+3d, \dots\)
\(n\)-th term:
\[ a_n = a + (n - 1)d. \]
Pehle \(n\) terms ka sum:
\[ S_n = \frac{n}{2}\big[\,2a + (n-1)d\,\big] = \frac{n}{2}\,(a + l), \]
jahan \(l = a_n\) last term hai.
AP \(3, 7, 11, \dots\) ka 10th term aur pehle 10 terms ka sum nikaalo. Yahan \(a = 3\), \(d = 4\). \[ a_{10} = 3 + 9(4) = 39, \qquad S_{10} = \frac{10}{2}(3 + 39) = 5 \times 42 = 210. \]
Arithmetic Mean (AM)
\(a\) aur \(b\) ka AM \(= \dfrac{a + b}{2}\). Ye dono ke beech ka aisa number hai jo unke saath AP banaye.
3. Geometric Progression (GP)
GP me har consecutive terms ka ratio constant hota hai, jise common ratio \(r\) kehte hain. First term \(a\).
GP aisi dikhti hai: \(a,\ ar,\ ar^{2},\ ar^{3}, \dots\)
\(n\)-th term:
\[ a_n = a\,r^{\,n-1}. \]
Pehle \(n\) terms ka sum (\(r \neq 1\)):
\[ S_n = \frac{a\,(r^{n} - 1)}{r - 1} = \frac{a\,(1 - r^{n})}{1 - r}. \]
GP \(2, 6, 18, \dots\) ke pehle 5 terms ka sum. Yahan \(a = 2\), \(r = 3\). \[ S_{5} = \frac{2(3^{5} - 1)}{3 - 1} = \frac{2(243 - 1)}{2} = 242. \]
Infinite GP ka sum
Agar \(|r| < 1\) ho, to infinite GP ka sum: \[ S_{\infty} = \frac{a}{1 - r}. \]
Geometric Mean (GM)
Positive \(a\) aur \(b\) ka GM \(= \sqrt{ab}\). Relation: kisi do positive numbers ke liye \(\text{AM} \ge \text{GM}\), yaani \(\dfrac{a+b}{2} \ge \sqrt{ab}\).
4. Kuch Special Sums
\[ \sum_{k=1}^{n} k = \frac{n(n+1)}{2} \]
\[ \sum_{k=1}^{n} k^{2} = \frac{n(n+1)(2n+1)}{6} \]
\[ \sum_{k=1}^{n} k^{3} = \left[\frac{n(n+1)}{2}\right]^{2} \]
Key Takeaways
- AP: constant difference \(d\); \(a_n = a + (n-1)d\), \(S_n = \frac{n}{2}(a + l)\).
- GP: constant ratio \(r\); \(a_n = ar^{n-1}\), \(S_n = \frac{a(r^{n}-1)}{r-1}\).
- Infinite GP (\(|r| < 1\)): \(S_\infty = \frac{a}{1-r}\).
- \(\text{AM} \ge \text{GM}\) for positive numbers.
- \(\sum k\), \(\sum k^{2}\), \(\sum k^{3}\) ke formulas yaad rakho.