Binomial Expansion, Sequences and Series

Question 1

Binomial Coefficient

meaning
terms
exponent of a
exponent of b

Answer 1

The coefficient of a term in a binomial expansion of the form (a + b)ⁿ

There are always n+1 terms for a binomial to the nᵗʰ power

Exponent of a decreases by 1 from left to right

Exponent of b increases by 1 from left to right

Question 2

Binomial theorem

formula
term bʳ
term aʳ

Answer 2

(nC0)aⁿb⁰ + (nC1)aⁿ⁻¹b¹ + (nC2)aⁿ⁻²b² + … + (nCn)a⁰bⁿ

(nCr)aⁿ⁻ʳbʳ

[nC(n-r)]aʳbⁿ⁻ʳ

Question 3

Finding the coefficient of x given two brackets

Answer 3

Expand the exponential term using binomial theorem,
combine the terms in the brackets with components of the binomial expansion to get the required coefficient then simplify the term to get the coefficient

Question 4

Finding the value of the constant term independent of x

Answer 4

r is unknown and solve by using the binomial theorem of term b, use the variable values and equate to x⁰ hence to zero, use the r value and multiply by the nCr term to get the value of the constant

Question 5

Finding the nᵗʰ term in the expansion

Answer 5

n is unknown, expand normally binomially for b and leave answer in (nCr)(x)ⁿ⁻ᶜ

Question 6

Finding equality for an estimation question

Answer 6

Expand the term with the power corresponding to its approximate expansion, after expanding cancel out each of the power to then solve the equation algebraically to find x. Sub the x value into the given equation that has been binomially expanded.

Question 7

Types of sequences

Answer 7

A sequence is mapping from the natural numbers to real numbers

Finite sequence - fixed number of terms

Infinite sequence - An ongoing term

Use of natural numbers is important as we have first, second and third terms only and no terms in between

Question 8

Sequence notation

Answer 8

Each number in a sequence is called a term,

the first term can be given by T₁, t₁, t(1) or T(1)
the general term or nᵗʰ is written as Tₙ or tₙ

Question 9

Expression of sequences

Answer 9

Listing, listing the terms in order
** use rigid brackets for sequences

Formula for nᵗʰ term

Recursive function, where each term is defined or calculated based on the previous term

Graphing, sequence is illustrated on a graph with the number of the term (n) on the horizontal axis and the term itself (tₙ) on the vertical axis
** Do not joint the points together

Question 10

Sigma notation

Answer 10

n = 4
Σ 2n + 3
n = 1

Question 11

Series

Answer 11

Obtained by adding the terms of a sequence of numbers together and can be written in sigma notation, infinite or finite

Question 12

Arithmetic sequence

Answer 12

Is a sequence were each term is calculated by adding a fixed amount to the previous term,
The first term of an arithmetic sequence is a
The fixed amount or common difference is d
The common difference can be calculated by subtracting a term from the one following it

general term of arithmetic sequences;
tₙ = a + (n-1)d

Question 13

Arithmetic series

Answer 13

If terms of an arithmetic sequence are added together it forms an arithmetic series

2, 4, 6, 8 - finite
2, 4, 6, 8, … - infinite

2 + 4 + 6 + 8 - arithmetic sequence
2 + 4 + 6 + 8 + … - arithmetic series

Sum of the first n terms of an arithmetic series

Sₙ = 0.5n (2a + [n-1]d)

Equivalent formula using the last term, l
Sₙ = 0.5n[a + l]

tₙ = Sₙ - Sₙ₋₁

Question 14

Geometric sequence

Answer 14

A sequence where each term is calculated by multiplying the previous term by a fixed number

t₁ = a
t₂ = ar
t₃ = ar²
t₄ = ar³

a is the first term of a sequence
common ratio is r, can be positive, negative an integer or fraction and is calculated by dividing any term by the one before it

r = tₙ₊₁ / tₙ

General term:

tₙ = arⁿ⁻¹

Question 15

Geometric series

Answer 15

Terms of a geometric sequence added together make a geometric series

(2, 4, 8, 16)* is a finite geometric sequence
* rigid brackets
2 + 4 + 8 + 16 is a finite geometric series

(2, 4, 8, 16, …)* is a infinite geometric sequence
* rigid brackets
2 + 4 + 8 + 16 + … is a infinite geometric series

Sₙ = [ a(rⁿ - 1) ] / [ r -1 ] where r > 1
and
Sₙ = [ a(1 - rⁿ) ] / [ 1 - r ] where r > 1

Question 16

Infinite Geometric series

Answer 16

the limit of a sequence is the number that the terms in the sequence when summed becomes very close to as n tends to a large number

Question 17

Sum to infinity

Answer 17

S∞ = a / r-1 where -1 < r < 1

Question 18

Sum to infinity exact fraction questions

Answer 18

convert the recurring decimal to a fraction to find the first term a then the common ratio r and then sub into the formula to solve, if given a decimal beginning with constant values, convert the constant value into a fraction then add it to the sum to infinity solved without the constant values