Binomial expansion

The first three terms, in ascending powers of x, of the binomial expansion

where n is a constant.

State the range of values of x for which the expansion is valid.

Circle your answer.

The first four terms in ascending powers of x of the binomial expansion of

are given by

where p and q are constants.

Find the value of p and the value of q

Hence find an approximation for 1.94⁵

Give your answer to five decimal places.

Find the coefficient of x⁷ in the expansion of (2x 3)⁷

Circle your answer.

Find the first three terms, in ascending powers of x, of the binomial expansion of

A student substitutes x 2 into the expansion of (1 x) to find an approximation

Explain the mistake in the student’s approach.

Give your answer to three significant figures.

Find the first two terms, in ascending powers of x, of the binomial expansion of

Hence, for small values of x, show that

where A, B and C are constants to be found.

The binomial expansion of (2 5x)⁴ is given by

4 2 3 4

Find the value of A and the value of B.

Show that

4 4 3

where C and D are constants to be found.

Hence, or otherwise, find

State the range of values of x for which the binomial expansion of

is valid.

Circle your answer.

Show that the first three terms, in descending powers of x, of the expansion of

are given by

10 9 8

where p and q are integers to be found.

Find the constant term in the expansion of

Show that the equation

simplifies to

Hence, solve the equation

Show that the first two terms of the binomial expansion of 4 2x² are

State the range of values of x for which the expansion found in part (a) is valid.

Hence, find an approximation for

giving your answer to five decimal places.

Fully justify your answer.

A student decides to use this method to find an approximation for

Explain why this may not be a suitable method.

Turn over for Section B

Section B

all questions in the spaces provided.

Circle your answer.

Find the first three terms, in ascending powers of x, of the binomial expansion

Hence, find the first three terms of the binomial expansion of

to seven decimal places.

Edward, a student, decides to use this method to find a more accurate value for the integral by increasing the number of terms of the binomial expansion used.

Explain clearly whether Edward’s approximation will be an overestimate, an underestimate, or if it is impossible to tell.

Edward goes on to use the expansion from part (b) to find an approximation

Explain why Edward’s approximation is invalid.

Find the coefficient of x² in the expansion of (1 2x)⁷

Circle your answer.

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