Greatest Terms for Positive Integral Index

In (a + b)ⁿ, let 'a' and 'b' be both positive numbers.

As r increases, the factor decreases. So long as this factor is greater than 1, T_r+1 remains greater than T_r. As r increases, cannot be always greater than 1.

In the beginning T_r+1 increases and after a certain stage, it starts decreasing.

T_r+1 is the greatest term where r is the greatest integer satisfying

Note 1:

In case  is an integer, then T_r+1 = T_r and both T_r+1 and T_r are the greatest terms.

Note 2:

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.

Working rules for finding the greatest term:

Step 1:

In (a + b)ⁿ, the constants a and b must be positive.

Step 3:

Simplify the inequality  and find the greatest possible value of r satisfy this inequality.

Step 4:

Calculate T_r+1 for this value of r. This gives the greatest term.

Example:

Find the greatest term in the expansion of (3 + 2x)⁹ when x = 1.

Suggested answer:

In (3 + 2x)⁹, we have

The greatest possible value of r is 4.

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Note:

When r = 4, we have T_{4 + 1} = T₄ .

\ T₅ and T₄ are both greatest terms.