Page 3 - Bigyan_Katha_Sepember_2020

‡m‡Þ¤^i 2020 3
gyL¨ cÖeÜ
MwY‡Z cÖMwZi aviYv I fviZxq Ae`vb
f¨Zvi Avw`Kvj ‡_‡K hvÎv ïiæ K‡i MwYZ bvbv kvLv-cÖkvLvq f‡i wM‡q AvR GK wekvj gnxiæ‡n
n mv‡c‡ÿ GK MvwYwZK weew„ Z, Ges m GKwU wbw`ó© ¯v^fvweKmsL¨v|hw`P(n)Ggbnq‡h, (i) P(1) mZ¨, Ges
(ii) P(m) mZ¨ a‡i Zvi ‡_‡K P(m + 1)- Gi mZ¨Zv cÖgvb Kiv hvq,
Zvn‡j P(n) wee„wZwU, ‡h ‡Kv‡bv ¯^vfvweK msL¨vi Rb¨ mZ¨ n‡e|
Gev‡i Dccv`¨wUi cÖgvb|
GLv‡b P(n) : tn = a + (n – 1)d
n = 1 ‡bIqv n‡j t1 = a, hv mZ¨| AZGe, P(1) mZ¨|
awi, P(m) mZ¨|
A_©vr, tm = a + (m – 1)d... .... (3) Zvn‡j tm+1 = tm + d = a + (m – 1)d + d = a + (m + 1 – 1)d
d‡j ‡`Lv ‡Mj P(m) mZ¨ n‡j P(m + 1)mZ¨| ∴ MvwYwZK Av‡ivn bxwZ Abymv‡i Dccv`¨wU cÖgvwYZ|
Gevi mgvšÍi cÖMwZi cÖ_g n-msL¨K
c‡`i mgwói m~Î wb‡q Av‡jvPbv| GKwU
mgvšÍi cÖMwZi cÖ_g c` a, mvaviY AšÍi d
Ges cÖ_g n-msL¨K c‡`i mgwó‡K Sn Øviv
m~wPZ Ki‡j Avgiv cv‡ev,
Sn = a + {a + d} + {a + 2d} + {a + 3d}
+ ... + {a + (n – 1)d}... .... (4)
Z¨evPx mi
m
Rb¨t =t =1,t =t +t ,hLbn≥3| 12 nn–1n–2
(vi)-Gi‡ÿ‡Îm‡~Îimvnv‡h¨cKÖ vkKivm¤¢e bv n‡jI, Zv wbqg ‡g‡bB wbav© wiZ; G‡ÿ‡Î tn n‡jv n-Zg ‡gŠwjK msL¨v|
cwiYZ n‡q‡Q| MwY‡Zi weKv‡ki m‡1⁄2 weKwkZ n‡q‡Q MwY‡Zi bvbv kvLv| GK GKwU kvLv mg„× n‡q‡Q bvbv MvwYwZK aviYv, wewfbœ c`, bvbv m~Î I Dccv`¨‡K wfwË K‡i| we`¨vjq MwY‡Zi exRMwYZ kvLvi GKwU ¸iæZ¡c~Y© aviYv n‡jv Abyμg (sequence) ev cÖMwZ (progression)-i aviYv| cvV¨eB‡Z GBme wb‡q cwi‡ewkZ ZË¡ I Z_¨ we‡kølY Ki‡j wKQy ‡ÿ‡Î wkÿvg~jK (pedagogic) `„wó‡KvY ‡_‡K wKQy ÎywU cwijwÿZ nq| ZvQvov GB aviYvi weKv‡ki HwZnvwmK ‡cÖÿvcU cÖvq Aby3 _v‡K| wKš‘ h_v_© wkÿ‡Yi ‡ÿ‡Î Gi ¸iæZ¡ Acwimxg| Dciš‘ GB aviYvi weKv‡k fviZxq‡`i Ae`vb kÖ×vi m‡1⁄2 ¯§iYxq|
Avgv‡`i Av‡jvPbvi cÖ_g As‡k _vK‡e cÖMwZ msμvšÍ MvwYwZK Z_¨ I Z‡Ë¡i weeiY Ges wØZxq As‡k _vK‡e Gi weKv‡ki BwZnvm I fviZxq Ae`vb|
μgvš^‡q mvRv‡bv GKwU msL¨v¸‡”Qi ‡h‡Kv‡bv ¯’v‡b Aew¯’Z GKwU msL¨v‡K GKwU we‡kl wbqgvbymv‡i wba©viY Kiv m¤¢e n‡j Zv‡K Abyμg ev cÖMwZ ejv nq| Gi cÖ_g msL¨v‡K Abyμ‡gi cÖ_g c` (t1) Ges n-Zg ¯’v‡b Aew¯’Z msL¨v‡K n-Zg c` (tn) ev mvaviY c` ejv nq| wb‡P Abyμ‡gi wKQy D`vniY ‡`Iqv n‡jv—
(i) 2, 5, 8, 11, 14, 17, ...
(ii) 2, 4, 8, 16, 32, 64, ...
(iii) 1, 3, 6, 10, 15, 21, ...
(iv) 1, 4, 9, 16, 25, 36
(v) 1, 1, 2, 3, 5, 8, 13, ...
(vi) 2, 3, 5, 7, 11, 13, 17, ...
g‡bivL‡Zn‡eGKwUAbμy ‡gimvaviYc` we‡kl ‡Kv‡bv wbqg ‡g‡b wbav© wiZ nq| GLv‡b ‡Zv (vi) Qvov evwK¸wji tn mwy bw`ó© m‡~ Îi Øviv cKÖ vk Kiv hv‡e| ‡hgb, (i)-Gi ‡ÿ‡Î tn =
3n – 1; (ii)-Gi ‡ÿ‡Î tn = 2n; (iii)-Gi ‡ÿ‡Î t = n(n + 1); (iv)-Gi ‡ÿ‡Î t = n2; (v)-Gi
GKwU cÖMwZ mmxg ev Amxg msL¨K c` wewkó n‡Z cv‡i| D`vniY, (iv) n‡jv mmxg cÖMwZ; evwK¸wj Amxg cÖMwZ|
we`¨vjq MwY‡Z g~jZ mgvšÍi cÖMwZ (arithmetic progression) I ¸‡YvËi cÖMwZ (geometric progression)-i Ici ‡Rvi ‡`Iqv nq|
mgvšÍi cMÖ wZ n‡jv Ggb GKUv Abμy g hvi ‡h‡Kv‡bv c` ‡_‡K Zvi wVK Av‡Mi c` we‡qvM Ki‡j memgq GKwU aæa eK msL¨v cvIqv hv‡e| GBaæaeKmsL¨vwU‡KmgvšÍicMÖ wZimvaviY AšÍi (common difference) ejv nq| D`vniY (i) n‡jv GKwU mgvšÍi cMÖ wZi D`vniY hvi mvaviY AšÍi n‡jv 2 (= 5 – 3)| GKwU mgvšÍi cMÖ wZi c_Ö g c` a Ges mvaviY AšÍi d n‡j mgvšÍi cMÖ wZwU n‡e:
a, a + d, a + 2d, a + 3d, ... ... ... (1) A_©vr, a + (1 – 1)d, a + (2 – 1)d, a + (3 – 1)d, ... ... a + (n – 1)d, ... .... (2)
(2)-G cKÖ vwkZ mgvšÍi cMÖ wZwU jÿ¨ Ki‡j ‡`Lvhv‡”Q,cMÖ wZicwÖZwUc``wyUAs‡ki mgwó—c_Ö g Ask n‡jv a Avi wØZxq Ask n‡jv (c`msL¨v – 1)d| web¨v‡miB mv`k„ ¨
jÿ K‡i cMÖ wZwUi mvaviY c`‡K ‡jLv hv‡etn =a+(n–1)d,hLbc_Ö gc`aI mvaviY AšÍiK d| wKš‘ wkÿvgj~ K `w„ ó‡KvY ‡_‡K mv`‡„ k¨i Ici wbfi© K‡i ‡bIqv
wm×všÍ MvwYwZKfv‡e MnÖ Y‡hvM¨ n‡Z cv‡i
bv| wm×všÍ‡K MnÖ Y‡hvM¨ Ki‡Z n‡j Zv‡K Dccv‡`¨i gva¨‡g cKÖ vk Ki‡Z n‡e Ges cgÖ vb Ki‡Z n‡e| GLv‡b ‡mUv ‡`Lv‡bv n‡jv| Dccv`¨: GKwU mgvšÍi cÖMwZi cÖ_g c` a Ges mvaviY AšÍi d n‡j cÖMwZwUi n-Zg c`wU n‡e: tn = a + (n – 1)d|
Dccv`¨wUi cÖgvY MvwYwZK Av‡ivn bxwZ (Principle of Mathematical Induction) cÖ‡qvM K‡i Kiv n‡e| D‡jøL¨, MwY‡Z cÖgv‡Yi ‡ÿ‡Î GB bxwZwU GKwU kw3kvjx nvwZqvi| Gi wee„wZ n‡jv—
awi, n GKwU ¯v^ fvweK msL¨v, P(n) n‡jv
AviS-GimÎ~ n‡jv:S =n {2a+(n–1)d}
2
m~ÎwUi cÖgvY mnR| (4)-Gi Wvbcÿ‡K
nn wecixZμ‡g mvRv‡j Avgiv cv‡ev,
Sn = {a + (n – 1)d} + {a + (n – 2)d} + ... + a... .... (5)
(4) Ges (5) ‡hvM K‡i cvB,
2Sn = {2a + (n – 1)d} + {2a + (n – 1) d} + ... + {2a + (n – 1)d} [n msL¨K] AZGe, S = n {2a + (n – 1)d}
n2
mgwói GB m~ÎwU MvwYwZK Av‡ivn bxwZ
n2n
cÖ‡qvM K‡iI cÖgvb Kiv hv‡e|
Gev‡i ¸‡YvËi cMÖ wZ wb‡q Av‡jvPbv|
GKwU cMÖ wZ‡K ¸‡YvËi cMÖ wZ ejv n‡e hw` Gic_Ö gc`kb~ ¨bvnqGesc_Ö gc‡`ici cwÖZwUc`‡KwVKZvice~e©Zx©c`‡KGKwU aæaeKmsL¨vw`qv¸YK‡icvIqvhvq|aæaeK msL¨v‡K ¸‡YvËi cMÖ wZi mvaviY Abcy vZ (common ratio) ejv nq| D`vniY (ii) GKwU ¸‡YvËi cMÖ wZi D`vniY, hvi c_Ö g c` 2 Ges mvaviY Abcy vZ 2 (= 4/2 = 8/4)|
GKwU ¸‡YvËi cÖMwZi cÖ_g c` a Ges mvaviY AbycvZ r n‡j ¸‡YvËi cÖMwZwU n‡e
cÂg c„ôvq `Öóe¨|