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Expressions of sequences for OCR A-level Maths

Expressions of sequences

This page covers the following topics:

1. nth term of arithmetic sequences
2. nth term of quadratic sequences
3. nth term of cubic sequences
4. nth term of geometric sequences

To determine the nth term expression for an arithmetic sequence, determine the first term in the sequence and the common first difference. Substituting the values to aโ‚™ = aโ‚ + d(n โˆ’ 1) and simplifying the expression obtained gives the nth term of an arithmetic sequence. To find a specific term in an arithmetic sequence, the value of n can be substituted into the nth term expression.

nth term of arithmetic sequences

To determine the nth term expression for a quadratic sequence, determine the second common difference in the sequence to obtain the coefficient a. Then determine the value b by using one of the first differences and the value of a. Finally, the value of c can be determined by using the expression for any term, preferably the first one. Substituting the values to aโ‚™ = anยฒ + bn + c gives the nth term of a quadratic sequence. To find a specific term in a quadratic sequence, the value of n can be substituted into the nth term expression.

nth term of quadratic sequences

To determine the nth term expression for a quadratic sequence, determine the third common difference in the sequence to obtain the coefficient a. Then determine the value b by using one of the second differences and the value of a. Then the value c can be determined by using the first differences. Finally, the value of d can be determined by using the expression for any term, preferably the first one. Substituting the values to aโ‚™ = anยณ + bnยฒ + cn + d gives the nth term of a quadratic sequence. To find a specific term in a quadratic sequence, the value of n can be substituted into the nth term expression.

nth term of cubic sequences

To determine the nth term expression for a geometric sequence, determine the first term in the sequence and the common ratio. Substituting the values to aโ‚™ = aโ‚r^(n โˆ’ 1) and simplifying the expression obtained gives the nth term of a geometric sequence. To find a specific term in a geometric sequence, the value of n can be substituted into the nth term expression.

nth term of geometric sequences

1

Jack is collecting marbles he found around his school. The marbles he has every day for the first four days are given below. Given that the number of marbles he has on any given day follows a cubic sequence, find how many marbles Jack has on the tenth day.

The cubic sequence is given by {0, 3, 8, 17}.
The sequence of first differences is given by {3, 5, 9, ...}.
The sequence of second differences is given by {2, 4, ...}.
The common third difference is 2.
Using the expressions from the general form, 6a = 2, thus a = 1/3.
12 ร— (1/3) + 2b = 2, thus b = โˆ’1.
7 ร— (1/3) + 3 ร— (โˆ’1) + c = 3, thus c = 11/3.
1/3 โˆ’ 1 + 11/3 + d = 0, thus d = 3.
Substituting into the general form, aโ‚™ = nยณ/3 โˆ’ nยฒ + 11n/3 + 3.
For n = 10, 10ยณ/3 โˆ’ 10ยฒ + 11 ร— (10)/3 + 3 = 273.

273

Jack is collecting marbles he found around his school. The marbles he has every day for the first four days are given below. Given that the number of marbles he has on any given day follows a cubic sequence, find how many marbles Jack has on the tenth day.

2

Leslie is collecting lemons from her tree. The mass of the lemons she has every week follows a geometric sequence. Given that on the seventh day, her lemons have mass 18.225 kg and that the common ratio of the sequence is 1.5, calculate the mass of the lemons she has on the eleventh day.

Using the formula for the general term of a geometric sequence, 18.225 = aโ‚(1.5)โถ.
Therefore the first term is 1.6.
Using the same formula, aโ‚โ‚ = 1.6(1.5)ยนโฐ = 92.2640625 kg.

92.2641 kg

Leslie is collecting lemons from her tree. The mass of the lemons she has every week follows a geometric sequence. Given that on the seventh day, her lemons have mass 18.225 kg and that the common ratio of the sequence is 1.5, calculate the mass of the lemons she has on the eleventh day.

3

Anna gets a new Job as a waitress. The tips she gets on the job are given by the geometric sequence {1, 4, 16, ...}, where the values are given in ยฃs. Calculate the tips she will receive on the fifth day.

The common ratio is given by 4 รท 1 = 4.
Using the formula for the general term of a geometric sequence, aโ‚… = 1 ร— 4โด = 256.

ยฃ256

Anna gets a new Job as a waitress. The tips she gets on the job are given by the geometric sequence {1, 4, 16, ...}, where the values are given in ยฃs. Calculate the tips she will receive on the fifth day.

4

Define the nth term of the sequence aโ‚™ = {1, 3, 5, 7, 9, โ€ฆ}.

d = 3 โˆ’ 1 = 2
Using the general formula for the n-th term, aโ‚™ = 1 + 2(n โˆ’ 1) = 2n โˆ’ 1.

2n โˆ’ 1

Define the nth term of the sequence aโ‚™ = {1, 3, 5, 7, 9, โ€ฆ}.

5

Find the nth term, or general rule, of the sequence Uโ‚™ = {โˆ’9, โˆ’6, โˆ’1, 6, 15, โ€ฆ}.

The sequence of first differences is given by {3, 5, 7, 9, ...}.
The common second difference is given by 2.
Using the expressions from the nth term of a quadratic sequence, 2a = 2, thus a = 1.
3 ร— 1 + b = 3, thus b = 0.
1 + 0 + c = โˆ’9, so c = โˆ’10.
Substituting into the general formula for the n-th term, Uโ‚™ = nยฒ โˆ’ 10.

nยฒ โˆ’ 10

Find the nth term, or general rule, of the sequence Uโ‚™ = {โˆ’9, โˆ’6, โˆ’1, 6, 15, โ€ฆ}.

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