A quadratic sequence has the following properties: The second difference is 10. The first two terms are equal, i.e. $T_1 = T_2$. $T_1 + T_2 + T_3 = 28$. 3.1 Show that the general term of the sequence is $T_n = 5n^2 - 15n + 16$. 3.2 Is 216 a term in this sequence? Justify your answer with the necessary calculations.

NSC Mathematics Number patterns, sequences and series: A quadratic sequence has the fol

A quadratic sequence has the following properties: The second difference is 10 - NSC Mathematics - Question 3 - 2023 - Paper 1

Question 3

A quadratic sequence has the following properties: The second difference is 10. The first two terms are equal, i.e. $T_1 = T_2$. $T_1 + T_2 + T_3 = 28$. 3.1 Show t... show full transcript

Worked Solution & Example Answer:A quadratic sequence has the following properties: The second difference is 10 - NSC Mathematics - Question 3 - 2023 - Paper 1

Step 1

Show that the general term of the sequence is $T_n = 5n^2 - 15n + 16$.

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Answer

To show that the general term of the sequence is expressed as the quadratic equation $T_n = 5n^2 - 15n + 16$ , we need to use the properties of the quadratic sequence provided.

Second Difference: The second difference of a quadratic sequence is constant. According to the question, the second difference is 10. Therefore, we know that the leading coefficient $a$ of the quadratic $T_n = an^2 + bn + c$ must satisfy:

ightarrow a = 5$$

Equal Terms: For the first two terms to be equal, we have: $T_1 = T_2$ Substituting in the form: $5(1)^2 + b(1) + c = 5(2)^2 + b(2) + c$ This simplifies to:

ightarrow b = -15$$

Sum of Terms: Now, substituting $T_1$ , $T_2$ , and $T_3$ : $T_1 + T_2 + T_3 = (5(1)^2 - 15(1) + 16) + (5(2)^2 - 15(2) + 16) + (5(3)^2 - 15(3) + 16)$ This results in: $[5 - 15 + 16] + [20 - 30 + 16] + [45 - 45 + 16]\ = 6 + 6 + 16\ = 28$ Thus confirming that the expression for $T_n$ is correct:
$T_n = 5n^2 - 15n + 16.$

Step 2

Is 216 a term in this sequence? Justify your answer with the necessary calculations.

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Answer

To determine if 216 is a term in the sequence, we need to set up the equation:

$T_n = 5n^2 - 15n + 16 = 216$

Rearranging gives us:

ightarrow 5n^2 - 15n - 200 = 0$$

Dividing by 5 simplifies this to: $n^2 - 3n - 40 = 0$
Now we can factor this quadratic equation: $(n - 8)(n + 5) = 0$ Thus, we have: $n = 8$ (only the positive integer solutions are valid as we are looking for term positions)
As $n = 8$ is a positive integer, we conclude that: $T_8 = 5(8)^2 - 15(8) + 16 = 216$ Therefore, 216 is indeed a term in this sequence.

A quadratic sequence has the following properties: The second difference is 10 - NSC Mathematics - Question 3 - 2023 - Paper 1

Worked Solution & Example Answer:A quadratic sequence has the following properties: The second difference is 10 - NSC Mathematics - Question 3 - 2023 - Paper 1

Show that the general term of the sequence is $T_n = 5n^2 - 15n + 16$.

Is 216 a term in this sequence? Justify your answer with the necessary calculations.

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