{"id":22073,"date":"2023-09-01T16:00:52","date_gmt":"2023-09-01T10:30:52","guid":{"rendered":"https:\/\/apboardsolutions.in\/?p=22073"},"modified":"2023-09-01T17:28:32","modified_gmt":"2023-09-01T11:58:32","slug":"inter-2nd-year-maths-2a-binomial-theorem-formulas","status":"publish","type":"post","link":"https:\/\/apboardsolutions.in\/inter-2nd-year-maths-2a-binomial-theorem-formulas\/","title":{"rendered":"Inter 2nd Year Maths 2A Binomial Theorem Formulas"},"content":{"rendered":"

Use these Inter 2nd Year Maths 2A Formulas<\/a> PDF Chapter 6 Binomial Theorem to solve questions creatively.<\/p>\n

Intermediate 2nd Year Maths 2A Binomial Theorem Formulas<\/h2>\n

\u2192 Let n be a positive integer and x, a be real numbers then
\n(x + a)n<\/sup> = n<\/sup>C0<\/sub>. xn<\/sup>. a0<\/sup> + n<\/sup>C1<\/sub>. xn – 1<\/sup>. a1<\/sup> + n<\/sup>C2<\/sub>. xn – 2<\/sup>. a2<\/sup> + ……… + n<\/sup>Cr<\/sub>. xn – r<\/sup>. ar<\/sup> + ……… + n<\/sup>Cn<\/sub>. x0<\/sup>. an<\/sup> = \\(\\sum_{r=0}^{n}\\) n<\/sup>Cr<\/sub>.xn – r<\/sup>. ar<\/sup>. and (x – a)n<\/sup> – n<\/sup>C0<\/sub>. xn<\/sup> – (x)n<\/sup> – n<\/sup>C1<\/sub>. xn – 1<\/sup>a + n<\/sup>C2<\/sub> xn – 2<\/sup>. a2<\/sup> …….. + ( – 1)r<\/sup> n<\/sup>Cr<\/sub>. xn – r<\/sup>. ar<\/sup> + ……… + (- 1)n<\/sup> n<\/sup>Cn<\/sub>.an<\/sup><\/p>\n

\"Inter<\/p>\n

\u2192 The expansion of (x + a)n<\/sup> contains (n + 1) terms.<\/p>\n

\u2192 In the expansion, the coefficients n<\/sup>C0<\/sub>, n<\/sup>C1<\/sub>, n<\/sup>C2<\/sub>, ….. n<\/sup>Cn<\/sub> are called binomial coefficients and these are simply denoted by C0<\/sub>, C1<\/sub>, C2<\/sub>, …….. Cn<\/sub>,.<\/p>\n

\u2192 In the expansion, (r + 1)th<\/sup> term is called the general term. It is denoted by Tr + 1<\/sub>.
\n\u2234 Tr + 1<\/sub> = c<\/sup>Cr<\/sub>. xn – r<\/sup>. ar<\/sup>, (0 \u2264 r \u2264 n)<\/p>\n

\u2192 The number of terms in the expansion of (a + b + c)n<\/sup> = \\(\\frac{(n+1)(n+2)}{2}\\)<\/p>\n

\u2192 If n is even in the expansion of (x + a)n<\/sup>, the middle term = T\\(\\left(\\frac{n}{2}+1\\right)\\)<\/sub><\/p>\n

\u2192 If n is odd in the expansion of (x + a)n<\/sup>, it has two middle terms which are T\\(\\left(\\frac{n+1}{2}\\right)\\)<\/sub>, T\\(\\left(\\frac{n+3}{2}\\right)\\)<\/sub>.<\/p>\n

\u2192 If \\(\\frac{(n+1)|x|}{|x|+1}\\) = p, a positive integer then pth<\/sup> and (p + 1)th<\/sup> terms are the numerically greatest terms in the expansion of (1 + x)n<\/sup>.<\/p>\n

\"Inter<\/p>\n

\u2192 If \\(\\frac{(n+1)|x|}{|x|+1}\\) = P + F where p is a positive integer and 0 < F < 1 then (p + 1)th<\/sup> the numerically greatest term in the expansion of (1 + x)n<\/sup><\/p>\n

\u2192 C0<\/sub> + C1<\/sub> + C2<\/sub> + ………. + Cn<\/sub> = 2n<\/sup><\/p>\n

\u2192 C0<\/sub> – C1<\/sub> + C2<\/sub> – C3<\/sub> + ……… + (- 1)n<\/sup>Cn<\/sub> = 0<\/p>\n

\u2192 C0<\/sub> + C2<\/sub> + C 4<\/sub> + …………… = C1<\/sub> + C3<\/sub> + C5<\/sub> + …………….. = 2n – 1<\/sup><\/p>\n

\u2192 \\(\\sum_{r=0}^{n}\\) n<\/sup>Cr<\/sub> = 2n<\/sup><\/p>\n

\u2192 \\(\\sum_{r=0}^{n}\\) r. n<\/sup>Cr<\/sub> = n. 2n – 1<\/sup><\/p>\n

\u2192 \\(\\sum_{r=2}^{n}\\) r(r – 1). n<\/sup>Cr<\/sub> = n(n – 1). 2n – 2<\/sup><\/p>\n

\u2192 \\(\\sum_{r=1}^{n}\\) r2<\/sup> . n<\/sup>Cr<\/sub> = n(n + 1). 2n – 2<\/sup><\/p>\n

\u2192 a. C0<\/sub> + (a + d). C1<\/sub> + (a + 2d). C2<\/sub> + ……… + (a + nd). Cn<\/sub> = (2a + nd) 2n – 1<\/sup><\/p>\n

\u2192 C0<\/sub>Cr<\/sub> + C1<\/sub>Cr + 1<\/sub> + C2<\/sub>Cr + 2<\/sub> + ………… + Cn – r<\/sub>. Cn<\/sub> = 2n<\/sup>Cn + r<\/sub><\/p>\n

\u2192 If f(x) = (a0<\/sub> + a1<\/sub>x + a2<\/sub>x2<\/sup> + ……… am<\/sub>xm<\/sup>)n<\/sup> then<\/p>\n