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Here are the first six terms of a quadratic sequence 10 19 34 55 82 115 Find an expression, in terms of n, for the nth term of this sequence.
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Here are the first six terms of a quadratic sequence 10 19 34 55 82 115 Find an expression, in terms of n, for the nth term of this sequence.
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Aug 6, 2024
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Here are the first six terms of a quadratic sequence
10 19 34 55 82 115
Find an expression, in terms of n, for the nth term of this sequence.
Mathematics
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Trichner
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Answer:
nth term = 3n² + 7
Explanation:
We can represent the nth term of a quadratic sequence as
an² + bn + c where a, b and c are constants
Plug in the given values for the first second and third terms (n = 1, 2 and 3) and solve for a, b and c to get the general expression for the nth term
For n = 1,
an² + bn + c = a(1²) + (1)b + c
=> a + b + c = 10 [1]
For n = 2,
an² + bn + c = a(2²) + (2)b + c = 19
=> 4a + 2b + c = 19 [2]
Subtract [1] from [2] to get
4a + 2b + c - (a + b + c) = 19 - 10
=> 4a + 2b + c - a - b - c = 9
=> 4a - a + 2b - b + c - c = 9
=> 3a + b = 9 [3]
For n = 3,
an² + bn + c = a(3²) + (3)b + c
=> 9a + 3b + c = 34
9a + 3b = 3(3a + b) by factoring out 3
So
9a + 3b + c = 34 becomes
3(3a + b) + c = 34
From eq 3 we have 3a + b = 9
Substituting for 3a + b we get
3(9) + c = 34
=> 27 + c = 24
=> c = 34 - 27= 7
Plug this value of c into equation [1], a + b + c = 10 to get
=> a + b + 7 = 10
=> a + b = 10 - 7
=> a + b = 3 [4]
We have equations 3 and 4 as
3a + b = 9 [3]
a + b = 3 [4]
Subtract equation [4] from [3] to get
[3] - [4]
=> 3a + b - (a + b) = 9 - 3
=> 2a + 0 = 6
=> a = 6/2 = 3
Plug the values of a = 3 and c = 7 into equation 1 to solve for b
a + b + c = 10
3 + b + 7 = 10
10 + b = 10
b = 0
So the quadratic sequence nth term is given by the expression
3n² + 0n + 7
which simplifies to
3n² + 7
Chulian
answered
Aug 13, 2024
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Chulian
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