QUESTION 2 [7] The sequence $$ 4 ; 9 ; x ; 37 ; \ldots $$ is a quadratic sequence. 2.1 Calculate $$ x $$. (3) 2.2 Hence, or otherwise, determine the $$ n^{ ext {th }} $$ term of the sequence. (4)

Question

QUESTION 2 [7] The sequence $$ 4 ; 9 ; x ; 37 ; \ldots $$ is a quadratic sequence. 2.1 Calculate $$ x $$. (3) 2.2 Hence, or otherwise, determine the $$ n^{	ext {th }} $$ term of the sequence. (4)

UpStudy AI · Accepted Answer

**Step 1. Calculate $$ x $$ using the constant second difference property**

Since the sequence is quadratic, the second differences are constant. Let the given terms be:
$$
a_1 = 4,\quad a_2 = 9,\quad a_3 = x,\quad a_4 = 37.
$$

First, compute the first differences:
$$
d_1 = a_2 - a_1 = 9 - 4 = 5,
$$
$$
d_2 = a_3 - a_2 = x - 9,
$$
$$
d_3 = a_4 - a_3 = 37 - x.
$$

For a quadratic sequence, the second differences are equal:
$$
d_2 - d_1 = d_3 - d_2.
$$

Substitute the known values:
$$
(x - 9) - 5 = (37 - x) - (x - 9).
$$

Simplify both sides:
$$
x - 14 = 37 - x - x + 9 = 46 - 2x.
$$

Solve for $$ x $$:
$$
x - 14 = 46 - 2x,
$$
$$
x + 2x = 46 + 14,
$$
$$
3x = 60,
$$
$$
x = \frac{60}{3} = 20.
$$

**Step 2. Determine the $$ n^\text{th} $$ term of the sequence**

Assume the $$ n^\text{th} $$ term has the form:
$$
a_n = An^2 + Bn + C.
$$

We have:
$$
a_1 = A(1)^2 + B(1) + C = A + B + C = 4,
$$
$$
a_2 = A(2)^2 + B(2) + C = 4A + 2B + C = 9,
$$
$$
a_3 = A(3)^2 + B(3) + C = 9A + 3B + C = 20.
$$

Now, set up the equations:
1. $$ A + B + C = 4 $$,
2. $$ 4A + 2B + C = 9 $$,
3. $$ 9A + 3B + C = 20 $$.

Subtract the first equation from the second:
$$
(4A + 2B + C) - (A + B + C) = 9 - 4,
$$
$$
3A + B = 5. \quad \text{(Equation 1)}
$$

Subtract the second equation from the third:
$$
(9A + 3B + C) - (4A + 2B + C) = 20 - 9,
$$
$$
5A + B = 11. \quad \text{(Equation 2)}
$$

Subtract Equation 1 from Equation 2:
$$
(5A + B) - (3A + B) = 11 - 5,
$$
$$
2A = 6,
$$
$$
A = 3.
$$

Substitute $$ A = 3 $$ into Equation 1:
$$
3(3) + B = 5,
$$
$$
9 + B = 5,
$$
$$
B = 5 - 9 = -4.
$$

Now, substitute $$ A = 3 $$ and $$ B = -4 $$ into the first equation:
$$
3 - 4 + C = 4,
$$
$$
-1 + C = 4,
$$
$$
C = 4 + 1 = 5.
$$

Thus, the $$ n^\text{th} $$ term is:
$$
a_n = 3n^2 - 4n + 5.
$$

**Final Answers:**

- $$ x = 20 $$.
- The $$ n^\text{th} $$ term of the sequence is $$ a_n = 3n^2 - 4n + 5 $$.
$$ x = 20 $$ and the $$ n^\text{th} $$ term of the sequence is $$ a_n = 3n^2 - 4n + 5 $$.

QUESTION 2 [7] The sequence \( 4 ; 9 ; x ; 37 ; \ldots \) is a quadratic sequence. 2.1 Calculate \( x \). (3) 2.2 Hence, or otherwise, determine the \( n^{\text {th }} \) term of the sequence. (4)

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Algebra Questions