Question

list the next three terms in the following sequence:
1, 9, 17, 25, 33, 41,...
also list the number that would be the twentieth number in the sequence
a. 49, 57, 65; 153
b. 50, 58, 64; 152
c. 49, 57, 65; 152
d. 50, 58, 64, 153
please select the best answer from the choices provided
a
b
c

Explanation:

Step1: Identify Sequence Pattern

The sequence is \(1, 9, 17, 25, 33, 41, \dots\). Calculate the difference between consecutive terms: \(9 - 1 = 8\), \(17 - 9 = 8\), \(25 - 17 = 8\), etc. So it's an arithmetic sequence with common difference \(d = 8\) and first term \(a_1 = 1\).

Step2: Find Next Three Terms

• 7th term: \(a_7 = a_6 + 8 = 41 + 8 = 49\)
• 8th term: \(a_8 = 49 + 8 = 57\)
• 9th term: \(a_9 = 57 + 8 = 65\)

Step3: Find 20th Term

Use the arithmetic sequence formula \(a_n = a_1 + (n - 1)d\). Here, \(a_1 = 1\), \(d = 8\), \(n = 20\).
\(a_{20} = 1 + (20 - 1) \times 8 = 1 + 19 \times 8 = 1 + 152 = 153\)? Wait, no—wait, \(a_1 = 1\), \(a_2 = 9 = 1 + 8(1)\), \(a_3 = 17 = 1 + 8(2)\), so \(a_n = 1 + 8(n - 1) = 8n - 7\). For \(n = 20\): \(8(20) - 7 = 160 - 7 = 153\)? Wait, but let's check the initial terms: \(n = 1\): \(8(1) - 7 = 1\) (correct), \(n = 2\): \(16 - 7 = 9\) (correct). Wait, but the options have 153 or 152. Wait, maybe I made a mistake. Wait, the first term is 1, so \(a_n = 1 + 8(n - 1) = 8n - 7\). For \(n = 20\): \(8*20 -7 = 160 -7 = 153\). But let's check the options. Option a: 49,57,65;153. Wait, but let's re - calculate the formula. Wait, \(a_1 = 1\), \(d = 8\), so \(a_n = a_1+(n - 1)d=1 + 8(n - 1)=8n - 7\). For \(n = 20\), \(8*20-7 = 153\). And the next three terms after 41 are 49,57,65. So the correct option should have next three terms 49,57,65 and 20th term 153, which is option a? Wait, but wait, let's check the options again. Option a: 49,57,65;153. Option c: 49,57,65;152. Wait, where is the mistake? Wait, \(a_1 = 1\), \(n = 1\): 1, \(n = 2\): 9, \(n = 3\):17, \(n = 4\):25, \(n = 5\):33, \(n = 6\):41, \(n = 7\):49, \(n = 8\):57, \(n = 9\):65. Then \(a_{20}=1+(20 - 1)*8=1 + 152 = 153\). So the next three terms are 49,57,65 and 20th term is 153, which is option a.

Answer:

a. 49, 57, 65; 153