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Introduction |
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A binomial is an algebraic expression of two terms which are connected by the operations '+' or '-'. |
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For n = 1,2,3,4, the expansion of (a + b)n, has been expressed in a very systematical manner in terms of combinatorial coefficients. The above expression suggest the conjecture that (a + b)n should be expressible in the form
for every natural number n. |
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Principle of Mathematical Induction |
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If P(n) is a statement (nÎN); such that |
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1. P(1) is true and
2. truth of P(k) implies the truth of P(k+1), then by the principle of
mathematical induction (P.M.I.), the statement P(n) is true for nÎN. |
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Alternative Proof of Binomial Theorem for Positive Integral Index (Combinatorial Method) |
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We have, (a + b)n = (a + b) (a + b) ....... n times. |
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The terms on the RHS are obtained by taking one letter from each factor and multiplying them together.
Choosing 'a' from all the factors, we get the term an. |
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Method of writing expansion |
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The first term in the expansion of (a + b)n is nC0anb0. For the second term, the coefficient is taken as nC1, the power of 'a' is decreased by one and the power of 'b' is increased by one. So, the second term is nC1an-1b1. For the third term, the coefficient is taken as nC2, the power of 'a' is again decreased by one and the power of 'b' is increased by one. This process goes on, till we get the last term as nCna0bn. |
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Some particular expansions for Positive Integral Index |
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Working rules for expanding (a + b)nnÎN: |
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Step 1: The value of index, n implies that there will be n+1 terms in the expansion of (a + b)n. |
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Step 2: Write the first term: nC0anb0. |
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Step 3: For the second term, take coefficient as nC1, decreases the power of 'a' by 1 and increases the power of 'b' by 1. Thus,
the second term is nC1an-1b1. |
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Step 4: For the third term, take coefficient as nC2, power of 'a' as n-2 and power of 'b' as 2. Continue this process repeatedly till the last term nCna0bn is obtained. |
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Step 5: For evaluating nCr, it is useful to write nCr as nCn-r, if r > n/2. |
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General Term for Positive Integral Index |
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General Term for Positive Integral Index is:
For 0  r  n, we have Tr+1=nCran-rbr. |
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Middle Terms for Positive Integral Index |
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The number of terms in the expansion of (a + b)n depends on the index n. The index n is either even or odd. |
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Particular Terms for Positive Integral Index |
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Sometimes, a particular term satisfying certain conditions is required in the binomial expansion of the type (a + b)n. This can be done by expanding (a + b)n and then locating the required term. Generally this becomes a tedious task, specially when the index n is large. In such cases, we begin by evaluating the general term Tr+1 and then finding the value of r by assuming Tr+1 to be the required term |
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Greatest Terms for Positive Integral Index |
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Working rules for finding the greatest term: |
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Step 1: In (a + b)n, the constants a and b must be positive. |
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Step 2: Write Tr+1 and Tr and find the value of Tr+1/Tr. |
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Step 3:Simplify the inequality (Tr+1/Tr) greater than or equal to 1 and
find the greatest possible value of r satisfy this inequality |
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Step 4: Calculate Tr+1 for this value of r. This gives the greatest term. |
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Some Applications of Binomial Theorem for Positive Integral Index |
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nC0, nC1, ..... nCn are called binomial coefficients. nC0, nC2 nC4, ..... are called even binomial coefficients. nC1, nC3, nC5 .... are called odd binomial coefficients. |
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In case of no ambiguity, the binomial coefficients nC0, nC1, ..... nCn are written as C0, C1, ..... Cn. |
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Binomial Theorem for Fractional Index |
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For any rational number n,
We accept this expansion without proof. |
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Some particular expansions for Fractional Index |
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For n  Q, we have:
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General Term for Fractional Index |
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The General Term for Fractional Index is:  |
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Particular Terms for Fractional Index |
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Sometimes, a particular term satisfying certain conditions is required in the binomial expansion of the type (1+x)n. This can be done by expanding (1+x)n to certain terms and then locating the required term. Generally this becomes a tedious task. In such cases, we begin by evaluating the general term Tr+1 and then finding the value of r by assuming Tr+1 to be the required term. |
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Some Applications of Binomial Theorem for Fractional Index |
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If x be numerically so small that its square and higher powers may be neglected, then (1+x)n= 1+nx (approximately), because x2, x3, x4,…. are all approximately zero. |
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Summary |
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1. A sentence is called a statement if it can be adjudged as true or false. Every statement is a sentence, but a sentence may or may not be a statement. |
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2. A statement involving natural number n is generally denoted by P(n). |
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3. A binomial is an algebraic expression of two terms which are connected by the operations '+' or '-'.
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