Sequences and Series

Question 1

Write Σ_n=1⁴ a_n longhand.

Answer 1

Σ_n=1⁴ a_n = a₁ + a₂ + a₃ + a₄

Question 2

What is the basic formula for finding the n^th term of an arithmetic sequence?

Answer 2

The n^th term in an arithmetic sequence is given by

a_n = a₁ + (n - 1)d

Where a₁ is the first term in the sequence, d is the common difference, and n is the number of the term to find.

Question 3

What is the basic formula for finding the n^th term of a geometric sequence?

Answer 3

The n^th term in a geometric sequence is given by

a_n = ar^n-1

Where a is the first term in the sequence, r is the factor between the terms (called the common ratio), and n is the number of the term to find.

Question 4

What are the two variations on the formula for finding the sum of a given number of terms in an arithmetic sequence?

Answer 4

The following formula can be used to find the sum of a certain number of terms of an arithmetic sequence

S_n = n/2 [a + l]

or

S_n = n/2 [2a + (n - 1)d)]

where S_n is the sum of n terms (n^th partial sum),
a is the first term, and l is the last (n^th) term. n is the number of terms and d is the common difference.

Question 5

In general, if you have n different terms, how many different arrangments can they form?

Answer 5

The number of possible combinations of n number of terms is called the n-factorial.

The n-factorial written as n! can be found using the following:

n! = n x (n - 1) x (n - 2) x … x 2 x 1

Question 6

What does (3!) / (0!) evaluate to?

Answer 6

(3!) / (0!) = 6 / 1

= 6

Question 7

What does (5!) / ([5-2]! • 2!) evaluate to?

Answer 7

(5!) / ([5-2]! • 2!) = (5!) / (3! • 2!)

= (5x4x3x2x1) / ([3x2x1] • [2x1])

= (5x4) / (2x1)

= 20 / 2

= 10

Question 8

Given that n! = n x (n - 1)! evaluate the following:

n! / (n - 1)!

Answer 8

n! / (n - 1)! = [n x (n - 1)!] / (n - 1)!

= n

Question 9

What is the formula for finding the n^th triangle number?

Answer 9

x_n = n(n + 1) / 2

e.g. The fifth triangle number, x₅ = 5(5 + 1) / 2

= 15

Question 10

Simplify Σ_r=1ⁿ af(r)

Hint: it’s to do with the constant term.

Answer 10

Σ_r=1ⁿ af(r) = aΣ_r=1ⁿ f(r)

e.g. Σ_r=1³ 3r² = 3(1²) + 3(2²) + 3(3²)

= 3[1² + 2² + 3² ]

= 3Σ_r=1³ r²

= 90

Question 11

Simplify Σ_r=1ⁿ a

Answer 11

Σ_r=1ⁿ a = a + a + … + a

= na

Question 12

What are the two variations on the formula for a Geometric Series?

Answer 12

The following formulae can be used to find the sum of a certain number of terms of a geometric sequence

S_n = [a (rⁿ - 1)] / [r - 1]

used when r > 1

or

S_n = [a (1 - rⁿ)] / [1 - r]

used when r < 1

where S_n is the sum of n terms (nth partial sum),
a is the first term, r is the common ratio, and n is the number of terms.

Question 13

What is the formula for a Geometric Series when n→infinity ?

Answer 13

When n→infinity the formula for a geometric series becomes:

S_infinity = a / [1 - r]

Because:

If -1 < r < 1

and n is large

S_n = [a (1 - rⁿ)] / [1 - r]

= a / [1 - r]

Since rⁿ → 0, n → infinity