50+ Math worksheets for 12th Grade on Quizizz | Free & Printable - Free Printable

Let's solve each of these problems step by step.

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1. Find the common difference: $ a_2 = 6 $ and $ a_6 = -14 $

We are given two terms in an arithmetic sequence:

- $ a_2 = 6 $
- $ a_6 = -14 $

In an arithmetic sequence:
$$
a_n = a_1 + (n-1)d
$$

So:
$$
a_2 = a_1 + d = 6 \quad \text{(1)}
$$
$$
a_6 = a_1 + 5d = -14 \quad \text{(2)}
$$

Subtract equation (1) from (2):
$$
(a_1 + 5d) - (a_1 + d) = -14 - 6
$$
$$
4d = -20 \Rightarrow d = -5
$$

✔ Answer: A) -5

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2. Find the common ratio (multiplier): $ a_1 = 4 $ and $ a_4 = 256 $

This is a geometric sequence:
$$
a_n = a_1 \cdot r^{n-1}
$$

Given:
$$
a_4 = a_1 \cdot r^3 = 256
$$
$$
4 \cdot r^3 = 256 \Rightarrow r^3 = \frac{256}{4} = 64
\Rightarrow r = \sqrt[3]{64} = 4
$$

✔ Answer: C) 4

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3. Write the nth term formula: $ 5 + 11 + 17 + 23 + \ldots + 35 $

This is an arithmetic sequence:
- First term $ a = 5 $
- Common difference $ d = 11 - 5 = 6 $

The general formula for the $ n $th term:
$$
t(n) = a + (n-1)d = 5 + (n-1)(6)
$$

So:
$$
t(n) = 5 + 6(n - 1)
$$

✔ Answer: A) $ t(n) = 5 + 6(n - 1) $

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4. Write the nth term formula: $ 3.5 + 7 + 14 + 28 + 56 $

Check if it's geometric:
- $ 7 / 3.5 = 2 $
- $ 14 / 7 = 2 $
- $ 28 / 14 = 2 $
- $ 56 / 28 = 2 $

Yes, geometric with:
- $ a = 3.5 $
- $ r = 2 $

Formula:
$$
a_n = a \cdot r^{n-1} = 3.5 \cdot (2)^{n-1}
$$

✔ Answer: D) $ a_n = 3.5(2)^{n-1} $

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5. Find the sum: $ 3 + 8 + 13 + 18 + \ldots + 98 $

Arithmetic series:
- $ a = 3 $
- $ d = 5 $
- Last term $ l = 98 $

Find number of terms $ n $:
$$
a_n = a + (n-1)d = 98
\Rightarrow 3 + (n-1)(5) = 98
\Rightarrow (n-1)(5) = 95 \Rightarrow n-1 = 19 \Rightarrow n = 20
$$

Sum formula:
$$
S_n = \frac{n}{2}(a + l) = \frac{20}{2}(3 + 98) = 10 \cdot 101 = 1010
$$

✔ Answer: C) 1010

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6. Find the sum: $ 11 + 12 + 13 + \ldots + 40 $

Arithmetic series:
- $ a = 11 $
- $ l = 40 $
- $ d = 1 $

Number of terms:
$$
n = 40 - 11 + 1 = 30
$$

Sum:
$$
S_n = \frac{n}{2}(a + l) = \frac{30}{2}(11 + 40) = 15 \cdot 51 = 765
$$

✔ Answer: B) 765

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7. Find the sum: $ 90 + 84 + 78 + \ldots + 24 $

Arithmetic sequence:
- $ a = 90 $
- $ d = -6 $
- $ l = 24 $

Find number of terms:
$$
a_n = a + (n-1)d = 24
\Rightarrow 90 + (n-1)(-6) = 24
\Rightarrow -6(n-1) = 24 - 90 = -66
\Rightarrow n-1 = 11 \Rightarrow n = 12
$$

Sum:
$$
S_n = \frac{n}{2}(a + l) = \frac{12}{2}(90 + 24) = 6 \cdot 114 = 684
$$

Wait — but 684 is not among the options.

Let’s double-check.

Wait! The options aren't shown here. But let's confirm our work.

Sequence: 90, 84, 78, ..., 24

Common difference: $ d = -6 $

$$
a_n = 90 + (n-1)(-6) = 24
\Rightarrow -6(n-1) = -66 \Rightarrow n-1 = 11 \Rightarrow n = 12
$$

Sum:
$$
S_{12} = \frac{12}{2}(90 + 24) = 6 \times 114 = 684
$$

But since the options aren’t visible in your image, perhaps they’re missing? Or maybe there’s a typo?

Wait — looking back at your image, question 7 has no answer choices listed, so we can only compute the sum.

✔ Sum = 684

If you have options, please check them; otherwise, this is correct.

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✔ Final Answers:

1. A) -5
2. C) 4
3. A) $ t(n) = 5 + 6(n - 1) $
4. D) $ a_n = 3.5(2)^{n-1} $
5. C) 1010
6. B) 765
7. Sum = 684 (No options provided)

Let me know if you'd like help with the remaining questions (8–17), or if you want to verify anything further!