Question

two students start a club the first week of school. each member invites three new members the second week of school. each of these new members then invites three additional members the following week. if the pattern of each new member inviting three new members each week continues, how many members will be in the club after five weeks?
60 students
121 students
242 students
363 students

Explanation:

Step1: Identify the pattern (geometric series)

The number of members each week forms a geometric sequence where the first term \(a = 2\) (initial members) and the common ratio \(r = 4\)? Wait, no. Wait, first week: 2 members. Second week: each of the 2 invites 3, so new members are \(2\times3 = 6\), total members \(2 + 6=8\)? Wait, no, maybe I misread. Wait, the problem says "each member invites three new members". So first week: 2 members. Second week: each of the 2 invites 3, so new members \(2\times3 = 6\), total members \(2+6 = 8\). Third week: each of the 8 members invites 3, so new members \(8\times3 = 24\), total \(8 + 24=32\). Wait, but that seems like a geometric series with first term \(a = 2\) and ratio \(r = 4\)? Wait, no, let's re - express.

Wait, actually, the total number of members after \(n\) weeks can be modeled as a geometric series. Let's think again.

First week (\(n = 1\)): \(N_1=2\)

Second week (\(n = 2\)): Each of the 2 members invites 3 new members. So the number of new members is \(2\times3\), and the total number of members is \(N_2=N_1+N_1\times3=N_1\times(1 + 3)=2\times4 = 8\)

Third week (\(n = 3\)): Each of the \(N_2\) members invites 3 new members. So total members \(N_3=N_2+N_2\times3=N_2\times4=8\times4 = 32\)

Wait, but this is a geometric sequence with \(a = 2\) and \(r = 4\). The formula for the sum of a geometric series \(S_n=\frac{a(r^n - 1)}{r - 1}\) when \(r eq1\). Wait, but in our case, the number of members each week is \(N_n=2\times4^{n - 1}\)? Wait, no, when \(n = 1\), \(N_1 = 2=2\times4^{0}\); \(n = 2\), \(N_2=8 = 2\times4^{1}\); \(n = 3\), \(N_3 = 32=2\times4^{2}\). So the general formula is \(N_n=2\times4^{n - 1}\)

We need to find the number of members after 5 weeks. So \(n = 5\)

\(N_5=2\times4^{5 - 1}=2\times4^{4}\)

\(4^{4}=256\), so \(N_5=2\times256 = 512\)? Wait, that's not matching the options. So I must have made a mistake.

Wait, maybe the interpretation is wrong. Maybe the number of new members each week is added, and the total number of members is a geometric series where the first term \(a = 2\) and the common ratio \(r=3\), but with the initial term. Wait, let's re - examine the problem.

"Two students start a club the first week of school. Each member invites three new members the second week of school. Each of these new members then invites three additional members the following week. If the pattern of each new member inviting three new members each week continues, how many members will be in the club after five weeks?"

Wait, maybe the total number of members after \(n\) weeks is a geometric series where the first term \(a = 2\) and the ratio \(r = 3\), but the formula for the total number of members is \(S_n=\sum_{k = 0}^{n - 1}2\times3^{k}\)

This is a geometric series with \(a = 2\), \(r = 3\), and number of terms \(n\) (for 5 weeks, number of terms is 5). The sum of a geometric series \(S_n=\frac{a(r^n - 1)}{r - 1}\)

So for \(n = 5\), \(a = 2\), \(r = 3\)

\(S_5=\frac{2(3^5 - 1)}{3 - 1}=\frac{2(243 - 1)}{2}=\frac{2\times242}{2}=242\)

Ah! That's one of the options (242 students). Let's verify:

Week 1: 2 members

Week 2: Each of 2 members invites 3 new members. New members = \(2\times3 = 6\). Total members = \(2+6 = 8\)

Week 3: Each of 8 members invites 3 new members. New members=\(8\times3 = 24\). Total members=\(8 + 24=32\)

Week 4: Each of 32 members invites 3 new members. New members=\(32\times3 = 96\). Total members=\(32+96 = 128\)

Week 5: Each of 128 members invites 3 new members. New members=\(128\times3 = 384\). Wait, no, that's not matching. Wait, no, my series sum was wrong. Wait, the sum formula:

Wait, the first term \(a = 2\) (week 1), then week 2: \(2\times3\), week 3: \(2\times3\times3=2\times3^{2}\), week 4: \(2\times3^{3}\), week 5: \(2\times3^{4}\)

Wait, no, the total number of members after \(n\) weeks is the sum of the number of members in each week. Wait, no, the number of members in week 1: 2; week 2: \(2 + 2\times3=2(1 + 3)=2\times4\); week 3: \(2\times4+2\times4\times3=2\times4(1 + 3)=2\times4^{2}\); week 4: \(2\times4^{2}+2\times4^{2}\times3=2\times4^{2}(1 + 3)=2\times4^{3}\); week 5: \(2\times4^{4}\)

But \(2\times4^{4}=2\times256 = 512\), which is not in the options. So there is a misinterpretation.

Wait, the problem says "how many members will be in the club after five weeks". Maybe "after five weeks" means including the first week. Let's count the number of members in each week:

Week 1: 2

Week 2: \(2+2\times3 = 8\)

Week 3: \(8+8\times3=8\times4 = 32\)

Week 4: \(32+32\times3=32\times4 = 128\)

Week 5: \(128+128\times3=128\times4 = 512\). No, not matching.

Wait, the options include 242. Let's check the sum of a geometric series with \(a = 2\), \(r = 3\), and \(n = 5\) terms (weeks 1 - 5). The sum \(S=\frac{2(3^{5}-1)}{3 - 1}=\frac{2(243 - 1)}{2}=242\). Ah, maybe the problem considers the number of members added each week as a geometric series where the first term is 2 (week 1), then week 2: \(2\times3\), week 3: \(2\times3^{2}\), week 4: \(2\times3^{3}\), week 5: \(2\times3^{4}\), and we sum them up.

Wait, week 1: 2

Week 2: \(2\times3 = 6\), total so far: \(2 + 6=8\)

Week 3: \(6\times3 = 18\), total so far: \(8+18 = 26\). No, that's not right.

Wait, I think the correct model is that the number of members in the club follows a geometric sequence where each week the number of members is multiplied by 4 (since each member brings 3 new members, so the total number of members is the old number plus 3 times the old number, i.e., 4 times the old number). But when we calculated using the sum of a geometric series with \(a = 2\), \(r = 4\), the sum after 5 weeks is \(S_5=\frac{2(4^{5}-1)}{4 - 1}=\frac{2(1024 - 1)}{3}=\frac{2\times1023}{3}=682\), which is not in the options.

But the option 242 is obtained when we use \(a = 2\), \(r = 3\), and sum for 5 terms. Let's check the number of new members:

Week 1: 2 members

Week 2: new members \(2\times3 = 6\), total \(2 + 6=8\)

Week 3: new members \(8\times3 = 24\), total \(8+24 = 32\)

Week 4: new members \(32\times3 = 96\), total \(32 + 96=128\)

Week 5: new members \(128\times3 = 384\), total \(128+384 = 512\). No, that's not 242.

Wait, maybe the problem is considering the number of members as a geometric sequence where the first term \(a = 2\) and the common ratio \(r = 3\), and we want the 5 - th term of the sequence (not the sum). The formula for the \(n\) - th term of a geometric sequence is \(a_n=a\times r^{n - 1}\). For \(n = 5\), \(a = 2\), \(r = 3\), \(a_5=2\times3^{4}=2\times81 = 162\), which is not in the options.

Wait, the options are 60, 121, 242, 363. Let's check 242. 242 divided by 2 is 121. 121 is 11 squared. Wait, maybe the pattern is \(2\times(1 + 3+9 + 27+81)\). Let's calculate the sum inside the parentheses: \(1+3 = 4\), \(4 + 9=13\), \(13+27 = 40\), \(40 + 81=121\). Then \(2\times121 = 242\). Ah! So the sum of the geometric series with \(a = 1\) (the multiplier for the initial 2), \(r = 3\), and \(n = 5\) terms: \(\sum_{k = 0}^{4}3^{k}=\frac{3^{5}-1}{3 - 1}=\frac{243 - 1}{2}=121\). Then multiply by 2 (the initial number of members) gives \(2\times121 = 242\). So that's the correct approach. The initial 2 members, and then each "generation" of new members is a geometric series with ratio 3. The total number of members is \(2\times\sum_{k = 0}^{4}3^{k}\)

Step2: Calculate the sum of the geometric series

The sum of a geometric series \(\sum_{k = 0}^{n - 1}r^{k}=\frac{r^{n}-1}{r - 1}\) when \(r eq1\). Here, \(r = 3\), \(n = 5\) (since we have 5 weeks, the number of terms in the series for the multipliers is 5: \(k = 0\) to \(k = 4\))

\(\sum_{k = 0}^{4}3^{k}=\frac{3^{5}-1}{3 - 1}=\frac{243 - 1}{2}=\frac{242}{2}=121\)

Then, multiply by the initial number of members (2): \(2\times121 = 242\)

Answer:

242 students