Question

1, 2, 4, 7, 11, 16,...
also list the number that would be the twentieth number in the sequence.
a. 23, 31, 40; 200
b. 23, 31, 40; 191
c. 22, 28, 35; 185
d. 22, 29, 37; 191
please select the best answer from the choices provided
a
b
c

Explanation:

Step1: Analyze the sequence pattern

The given sequence is \(1, 2, 4, 7, 11, 16, \dots\). Let's find the differences between consecutive terms:

• \(2 - 1 = 1\)
• \(4 - 2 = 2\)
• \(7 - 4 = 3\)
• \(11 - 7 = 4\)
• \(16 - 11 = 5\)

So the differences between consecutive terms are increasing by 1 each time. The \(n\)-th term of the sequence can be found using the formula for the sum of the first \(n - 1\) positive integers plus the first term. The sum of the first \(k\) positive integers is \(\frac{k(k + 1)}{2}\), so the \(n\)-th term \(a_n\) of the sequence is \(a_n=1+\frac{(n - 1)n}{2}\).

Step2: Find the next few terms

• For the 7th term (\(n = 7\)): \(a_7=1+\frac{6\times7}{2}=1 + 21 = 22\)? Wait, no, wait. Wait, the 6th term is 16. The difference between 6th and 7th term should be 6. So 16+6 = 22? Wait, no, wait the differences are 1,2,3,4,5, so the next difference is 6, so 16 + 6=22? Wait, no, wait the 1st term: 1, 2nd:1 + 1=2, 3rd:2+2 = 4, 4th:4 + 3=7, 5th:7+4 = 11, 6th:11+5 = 16, 7th:16+6 = 22, 8th:22+7 = 29, 9th:29+8 = 37. Wait, but let's check the options. Option d has 22,29,37. Wait, but let's check the 20th term.

The formula for the \(n\)-th term of the sequence: The sequence of differences is \(1,2,3,\dots,(n - 1)\). So the \(n\)-th term \(a_n=1+\sum_{k = 1}^{n - 1}k\). Since \(\sum_{k=1}^{m}k=\frac{m(m + 1)}{2}\), so \(a_n=1+\frac{(n - 1)n}{2}\).

For \(n = 20\): \(a_{20}=1+\frac{19\times20}{2}=1+190 = 191\).

Now let's check the options:

• Option a: Next terms after 16 should be 23? But according to our calculation, 16+6=22, so a is wrong.
• Option b: Next term after 16 is 23? No, should be 22, so b is wrong.
• Option c: Next term after 16 is 22? Then 22+6=28? No, the difference after 6 should be 7, so 22+7=29, not 28. So c is wrong.
• Option d: Next terms: 16+6=22, 22+7=29, 29+8=37. And 20th term is 191. So d matches? Wait, wait, earlier when I calculated the 7th term, I thought 16+6=22 (7th term), 8th term:22+7=29, 9th term:29+8=37. Then let's check the formula for \(n = 7\): \(a_7=1+\frac{6\times7}{2}=1 + 21=22\), correct. \(n = 8\): \(1+\frac{7\times8}{2}=1 + 28 = 29\), correct. \(n = 9\): \(1+\frac{8\times9}{2}=1+36 = 37\), correct. \(n = 20\): \(1+\frac{19\times20}{2}=1 + 190=191\), correct. So option d is correct? Wait, but wait the original sequence is 1,2,4,7,11,16. Let's check the differences again: 2-1=1, 4-2=2, 7-4=3, 11-7=4, 16-11=5. So the differences are 1,2,3,4,5. So the next difference is 6, so 16+6=22 (7th term), then 22+7=29 (8th), 29+8=37 (9th). Then the 20th term: using the formula \(a_n=1+\frac{(n - 1)n}{2}\), for \(n = 20\), \(a_{20}=1+\frac{19\times20}{2}=1 + 190=191\). So option d has 22,29,37 and 191, which matches. Wait, but let's check the options again. Option d: 22,29,37; 191. Yes. So the correct answer is d? Wait, but wait the initial sequence: 1,2,4,7,11,16. Let's check the 7th term: 16 + 6=22 (correct, since difference is 6), 8th term:22 + 7=29 (difference 7), 9th term:29 + 8=37 (difference 8). Then 20th term: sum of differences from 1 to 19, plus 1. Sum from 1 to 19 is \(\frac{19\times20}{2}=190\), plus 1 is 191. So yes, option d is correct. Wait, but the options:

a. 23, 31, 40; 200

b. 23, 31, 40; 191

c. 22, 28, 35; 185

d. 22, 29, 37; 191

So the correct option is d. Wait, but earlier when I thought the 7th term is 22, 8th 29, 9th 37, and 20th 191, which matches option d. So the answer is d.

Answer:

d. 22, 29, 37; 191