Question

1. some terms in a sequence are given in the table below. is this sequence arithmetic, geometric or neither? how do you know? \\(\

$$\begin{array}{c|c|c|c|c|c|c|c} \\text{term number } (n) & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\ \\hline n^{\\text{th}} \\text{ term} & 2 & -4 & 6 & -8 & 10 & -12 & 14 \\end{array}$$

\\)

Explanation:

Step1: Check for arithmetic sequence

To check if it's arithmetic, find the difference between consecutive terms.
- Between term 1 (2) and term 2 (-4): $-4 - 2 = -6$
- Between term 2 (-4) and term 3 (6): $6 - (-4) = 10$
Wait, that's inconsistent? Wait, no, wait, let's recalculate. Wait, term 1: 2, term 2: -4. Difference: $-4 - 2 = -6$? Wait, no, arithmetic sequence has constant difference. Wait, maybe I made a mistake. Wait, term 2 - term 1: $-4 - 2 = -6$; term 3 - term 2: $6 - (-4) = 10$? That can't be. Wait, no, wait the sequence is 2, -4, 6, -8, 10, -12, 14. Wait, maybe the pattern is alternating sign and multiplying by 2? Wait, no, let's check the differences again. Wait, term 1: 2, term 2: -4. Difference: -4 - 2 = -6. Term 3: 6. Difference from term 2: 6 - (-4) = 10. That's not constant. Wait, but wait, maybe I misread the terms. Wait, term 1: 2, term 2: -4, term 3: 6, term 4: -8, term 5: 10, term 6: -12, term 7:14. Wait, let's check the absolute values: 2,4,6,8,10,12,14. The absolute values have a common difference of 2. And the sign alternates: positive, negative, positive, negative, etc. Wait, but for arithmetic sequence, the difference between consecutive terms should be constant. Wait, let's check the difference between term 3 and term 4: -8 - 6 = -14? No, wait, term 3 is 6, term 4 is -8. So -8 - 6 = -14? That's not right. Wait, no, I think I messed up the subtraction. Wait, term 2 is -4, term 3 is 6: 6 - (-4) = 10. Term 3 is 6, term 4 is -8: -8 - 6 = -14. That's not constant. Wait, but maybe the sequence is defined by a formula. Let's see the nth term. For n=1: 2 = 2*1*(-1)^(1+1) (since (-1)^2=1, 2*1*1=2). n=2: -4 = 2*2*(-1)^(2+1) = 4*(-1)^3 = -4. n=3: 6 = 2*3*(-1)^(3+1)=6*1=6. Yes! So the formula is $a_n = 2n(-1)^{n + 1}$. Now, let's check the difference between consecutive terms for arithmetic. Wait, maybe I made a mistake earlier. Let's recalculate the differences correctly. Term 1: 2, Term 2: -4. Difference: -4 - 2 = -6. Term 3: 6. Difference: 6 - (-4) = 10. Term 4: -8. Difference: -8 - 6 = -14. Term 5: 10. Difference: 10 - (-8) = 18. These differences are not constant, so not arithmetic? Wait, but wait, the absolute values: 2,4,6,8,10,12,14. The absolute values have a common difference of 2. But the signs alternate. Wait, maybe the sequence is arithmetic in absolute value but with alternating signs? No, arithmetic sequence requires constant difference between consecutive terms, regardless of sign. Wait, but let's check the ratio for geometric sequence. Term 2 / Term 1: -4 / 2 = -2. Term 3 / Term 2: 6 / (-4) = -1.5. Not constant. So geometric ratio is not constant. Wait, but wait, maybe I made a mistake in the arithmetic difference. Wait, let's list the terms:

n: 1, 2, 3, 4, 5, 6, 7

a_n: 2, -4, 6, -8, 10, -12, 14

Now, let's compute the difference between a_{n+1} and a_n:

a_2 - a_1 = -4 - 2 = -6

a_3 - a_2 = 6 - (-4) = 10

a_4 - a_3 = -8 - 6 = -14

a_5 - a_4 = 10 - (-8) = 18

a_6 - a_5 = -12 - 10 = -22

a_7 - a_6 = 14 - (-12) = 26

These differences are -6, 10, -14, 18, -22, 26. The differences between these differences (second difference) would be 16, -24, 32, -40, 48. Not constant. So not arithmetic.

Now check geometric: ratio between a_2 and a_1: -4/2 = -2

a_3/a_2: 6/(-4) = -1.5

Not the same, so not geometric.

Wait, but wait, the absolute values: 2,4,6,8,10,12,14. The difference between absolute values is 2 (4-2=2, 6-4=2, etc.). But the actual terms have alternating signs. So the sequence is 2, -4, 6, -8, 10, -12, 14. Let's check the difference between term 1 and term 2: -6, term 2 and term 3: +10, term 3 and term 4: -14, term 4 and term 5: +18, term 5 and term 6: -22, term 6 and term 7: +26. These differences are increasing by 4 each time? Wait, -6 to +10: difference is 16? No, -6, +10, -14, +18, -22, +26. The pattern here is that the difference between consecutive terms is -6, +10, -14, +18, -22, +26. Notice that the absolute value of the difference is 6, 10, 14, 18, 22, 26, which is an arithmetic sequence with common difference 4. But the original sequence's differences are not constant, so it's not arithmetic. And the ratios are not constant, so not geometric. Wait, but wait, maybe I made a mistake. Wait, let's check the formula again. The nth term is 2n*(-1)^(n+1). So for n=1: 2*1*1=2, n=2: 2*2*(-1)=-4, n=3: 2*3*1=6, n=4: 2*4*(-1)=-8, etc. So the sequence is defined by that formula. Now, to check if it's arithmetic, we need constant difference. Let's compute the difference between a_{n+1} and a_n:

a_{n+1} - a_n = 2(n+1)(-1)^{(n+1)+1} - 2n(-1)^{n+1}

Simplify:

= 2(n+1)(-1)^{n+2} - 2n(-1)^{n+1}

Note that (-1)^{n+2} = (-1)^{n+1}*(-1), so:

= 2(n+1)(-1)^{n+1}*(-1) - 2n(-1)^{n+1}

= -2(n+1)(-1)^{n+1} - 2n(-1)^{n+1}

= -2(-1)^{n+1}[ (n+1) + n ]

= -2(-1)^{n+1}(2n + 1)

This is not a constant, so the difference between consecutive terms is not constant. Therefore, it's not an arithmetic sequence.

For geometric sequence, the ratio a_{n+1}/a_n should be constant.

a_{n+1}/a_n = [2(n+1)(-1)^{n+2}] / [2n(-1)^{n+1}] = [(n+1)(-1)^{n+2}] / [n(-1)^{n+1}] = (n+1)/n * (-1)

This ratio depends on n (since (n+1)/n is not constant), so the ratio is not constant. Therefore, it's not a geometric sequence.

Wait, but the problem says "Is this sequence arithmetic, geometric, or neither?" So based on the above, it's neither? But wait, maybe I made a mistake. Wait, let's check the differences again with the actual terms:

Term 1: 2

Term 2: -4. Difference: -4 - 2 = -6

Term 3: 6. Difference: 6 - (-4) = 10

Term 4: -8. Difference: -8 - 6 = -14

Term 5: 10. Difference: 10 - (-8) = 18

Term 6: -12. Difference: -12 - 10 = -22

Term 7:14. Difference:14 - (-12) = 26

These differences (-6, 10, -14, 18, -22, 26) are not constant, so not arithmetic. Ratios:

-4/2 = -2

6/-4 = -1.5

-8/6 ≈ -1.333

10/-8 = -1.25

-12/10 = -1.2

14/-12 ≈ -1.166

Not constant, so not geometric. Therefore, the sequence is neither arithmetic nor geometric.

Wait, but wait, the absolute values form an arithmetic sequence (2,4,6,8,10,12,14) with common difference 2. But the original sequence has alternating signs, so the actual sequence's differences are not constant. So the answer is neither? But let's check again. Wait, maybe the problem has a typo, or I misread. Wait, the terms are 2, -4, 6, -8, 10, -12, 14. Let's check the difference between term 1 and term 2: -6, term 2 and term 3: +10, term 3 and term 4: -14, term 4 and term 5: +18, term 5 and term 6: -22, term 6 and term 7: +26. The pattern of the differences is that each difference is 4 more than the previous difference in absolute value, and alternating sign. So the differences are an arithmetic sequence with common difference 4, but the original sequence's differences are not constant, so it's not arithmetic. And ratios are not constant, so not geometric. So the answer is neither.

Wait, but maybe I made a mistake in the arithmetic difference. Let's check again:

Term 1: 2

Term 2: -4. 2 + (-6) = -4. So difference is -6.

Term 3: -4 + 10 = 6. Difference is +10.

Term 4: 6 + (-14) = -8. Difference is -14.

Term 5: -8 + 18 = 10. Difference is +18.

Term 6: 10 + (-22) = -12. Difference is -22.

Term 7: -12 + 26 = 14. Difference is +26.

So the differences are -6, +10, -14, +18, -22, +26. The absolute values of the differences: 6,10,14,18,22,26. The difference between these absolute values is 4 (10-6=4, 14-10=4, etc.). So the differences of the original sequence form an arithmetic sequence with common difference 4, but the original sequence itself does not have a constant difference, so it's not arithmetic. And the ratios are not constant, so not geometric. Therefore, the sequence is neither arithmetic nor geometric.

Answer:

The sequence is neither arithmetic nor geometric. To determine this:

• Arithmetic Check: Calculate differences between consecutive terms: \( -4 - 2 = -6 \), \( 6 - (-4) = 10 \), \( -8 - 6 = -14 \), etc. These differences (\( -6, 10, -14, \dots \)) are not constant.
• Geometric Check: Calculate ratios between consecutive terms: \( \frac{-4}{2} = -2 \), \( \frac{6}{-4} = -1.5 \), \( \frac{-8}{6} \approx -1.33 \), etc. These ratios are not constant.

Thus, the sequence is neither arithmetic nor geometric.