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| Greatest Terms for Positive Integral Index |
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| In (a + b)n, let 'a' and 'b' be both positive numbers. |
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As r increases, the factor
decreases. So long as this factor is greater than 1, Tr+1
remains greater than Tr. As r increases,
cannot be always greater than 1. |
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 In the beginning Tr+1 increases and after a certain stage, it starts decreasing. |
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 Tr+1 is the greatest term where r is the greatest integer satisfying  |
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| Note 1: |
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In case  is an integer, then Tr+1
= Tr and both Tr+1
and Tr are the greatest terms. |
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| Note 2: |
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| If a and b are positive numbers, then the above method can also be applied to find the numerically greatest term in the expansion of (a - b)n. |
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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 3: Simplify the inequality  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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| Example: |
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| Find the greatest term in the expansion of (3 + 2x)9 when x = 1. |
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| Suggested answer: |
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| In (3 + 2x)9, we have |
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 The greatest possible value of r is 4. |
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| = 489888 |
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| Note: |
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| When r = 4, we have T4 + 1 = T4 . |
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| \ T5 and T4 are both greatest terms. |
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