Maths-
General
Easy

Question

Find the third term of the binomial expansion left parenthesis a minus 3 right parenthesis to the power of 6 .

The correct answer is: Thus, the third term of the expansion is .


    ANSWER:
    Hint:
    The binomial expansion is left parenthesis x plus y right parenthesis to the power of n equals sum from k equals 0 to n of   n C subscript k x to the power of n minus k end exponent y to the power of k , here n greater or equal than 0.
    We are asked to find the third term of the binomial expansion left parenthesis a minus 3 right parenthesis to the power of 6 .
    Step 1 of 2:
    The given expression is left parenthesis a minus 3 right parenthesis to the power of 6 . Here, x equals a straight & y equals negative 3 . The value of n = 6 . Thus, there would be seven terms in the expansion.
    Step 2 of 2:
    Substitute the values of the expression in the binomial expansion to get its terms:
    left parenthesis a minus 3 right parenthesis to the power of 6 equals 6 C subscript 0 a to the power of 6 plus 6 C subscript 1 open parentheses a to the power of 5 close parentheses left parenthesis negative 3 right parenthesis plus 6 C subscript 2 open parentheses a to the power of 4 close parentheses left parenthesis negative 3 right parenthesis squared plus 6 C subscript 3 open parentheses a cubed close parentheses left parenthesis negative 3 right parenthesis cubed plus 6 C subscript 4 open parentheses a squared close parentheses left parenthesis negative 3 right parenthesis to the power of 4 plus 6 C subscript 5 left parenthesis a right parenthesis left parenthesis negative 3 right parenthesis to the power of 5 plus 6 C subscript 6 left parenthesis negative 3 right parenthesis to the power of 6
    equals a to the power of 6 plus 6 a to the power of 5 left parenthesis negative 3 right parenthesis plus 15 a to the power of 4 left parenthesis 9 right parenthesis plus 20 a cubed left parenthesis negative 27 right parenthesis plus 15 a squared left parenthesis 81 right parenthesis plus 6 a left parenthesis negative 243 right parenthesis plus 729
    equals a to the power of 6 minus 18 a to the power of 5 plus 135 a to the power of 4 minus 540 a cubed plus 1215 a squared minus 1458 a plus 729

    Thus, the third term of the expansion is135 a to the power of 4 .
    Note:
    The expansion of left parenthesis x plus y right parenthesis to the power of n has n+1 terms while expanding. The answer can be found using the Pascal’s triangle or binomial expansion.

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