Question

jackie plants 2 flowers in a planter box. every week, she plants one more flower in the planter box. how many flowers does jackie have in her planter box after 5 weeks? jackie has \boxed{} flowers in her planter box after 5 weeks.

Explanation:

Step1: Identify the sequence pattern

Jackie starts with 2 flowers. Then each week, she plants 1 more. So the number of flowers each week forms an arithmetic sequence: first term \( a_1 = 2 \), common difference \( d = 1 \), and we need the sum after \( n = 5 \) weeks (including the initial planting? Wait, actually, let's think in terms of weeks: Week 0 (initial): 2, Week 1: 3, Week 2: 4, Week 3: 5, Week 4: 6? Wait no, the problem says "after 5 weeks". Wait, initial planting is before any weeks? Wait, the problem says "Jackie plants 2 flowers in a planter box. Every week, she plants one more flower in the planter box. How many flowers does Jackie have in her planter box after 5 weeks?"

So let's list the number of flowers planted each week:

Week 1 (first week after initial): plants 3? Wait no, initial is 2, then every week (so week 1: 2 + 1 = 3, week 2: 3 + 1 = 4, week 3: 4 + 1 = 5, week 4: 5 + 1 = 6, week 5: 6 + 1 = 7? Wait no, maybe the initial is week 0, and we need the total after 5 weeks (so weeks 0 to 5, 6 terms? No, the problem says "after 5 weeks", so maybe the initial is week 0, and then 5 weeks pass, so weeks 0 (initial), week 1, week 2, week 3, week 4, week 5? Wait, no, maybe the initial planting is the start, and then each week after that she plants one more. So the number of flowers each week (including the initial) is:

Week 0 (start): 2

Week 1: 2 + 1 = 3

Week 2: 3 + 1 = 4

Week 3: 4 + 1 = 5

Week 4: 5 + 1 = 6

Week 5: 6 + 1 = 7

Wait, but that's 6 terms. But maybe the problem is that the initial 2 is week 0, and then after 5 weeks, we have week 0 to week 5, which is 6 weeks? No, maybe the initial planting is week 1, and then 5 weeks: week 1: 2, week 2: 3, week 3: 4, week 4: 5, week 5: 6. Wait, that's 5 weeks, but then the total would be the sum of an arithmetic sequence with first term 2, common difference 1, number of terms 5? Wait, no, let's check the problem again: "Jackie plants 2 flowers in a planter box. Every week, she plants one more flower in the planter box. How many flowers does Jackie have in her planter box after 5 weeks?"

So the number of flowers planted each week:

- Week 1 (first week): 2 flowers

- Week 2: 2 + 1 = 3 flowers

- Week 3: 3 + 1 = 4 flowers

- Week 4: 4 + 1 = 5 flowers

- Week 5: 5 + 1 = 6 flowers

Wait, no, that's 5 weeks, but the total number of flowers is the sum of these? Wait, no, maybe she plants 2 initially, then every week after that, she plants one more. So the total number of flowers after n weeks is the sum of the arithmetic series where the first term \( a_1 = 2 \), common difference \( d = 1 \), and number of terms \( n + 1 \) (since initial is week 0, then n weeks). Wait, after 5 weeks, the number of terms is 6 (week 0 to week 5). The formula for the sum of an arithmetic series is \( S_n = \frac{n}{2}(2a_1 + (n - 1)d) \). Wait, if n = 6 (terms), then \( S_6 = \frac{6}{2}(2*2 + (6 - 1)*1) = 3*(4 + 5) = 3*9 = 27 \)? No, that can't be right. Wait, maybe I'm misunderstanding the problem. Maybe "every week, she plants one more flower" means that each week, the number of flowers she plants increases by 1, but the total number of flowers is the sum of the number of flowers planted each week.

Wait, initial planting: 2 flowers (week 0)

Week 1: plants 2 + 1 = 3 flowers

Week 2: plants 3 + 1 = 4 flowers

Week 3: plants 4 + 1 = 5 flowers

Week 4: plants 5 + 1 = 6 flowers

Week 5: plants 6 + 1 = 7 flowers

Wait, no, that would be the number of flowers planted each week, but the total number of flowers is the sum of all these. Wait, no, maybe the problem is that she plants 2 flowers initially, and then each week, she adds one more to the planter box (i.e., the total number of flowers increases by 1 each week). Wait, that would be a linear sequence. Wait, initial: 2, week 1: 3, week 2: 4, week 3: 5, week 4: 6, week 5: 7. So after 5 weeks, the total number of flowers is 7? But that seems low. Wait, no, maybe the problem is that "every week, she plants one more flower" means that each week, the number of flowers she plants is 2, then 3, then 4, etc., and we need the total after 5 weeks (i.e., 5 plantings: week 1 to week 5). Wait, initial is week 0, then week 1 to week 5 is 5 weeks. So number of flowers planted each week:

Week 0: 2

Week 1: 3

Week 2: 4

Week 3: 5

Week 4: 6

Week 5: 7

Wait, that's 6 plantings, but the problem says "after 5 weeks", so maybe week 0 is initial, then 5 weeks (week 1 to week 5), so total plantings: 6. But the sum of 2 + 3 + 4 + 5 + 6 + 7. Let's calculate that: 2 + 3 = 5, 5 + 4 = 9, 9 + 5 = 14, 14 + 6 = 20, 20 + 7 = 27. No, that's too high. Wait, maybe the problem is that "every week, she plants one more flower" means that each week, the number of flowers in the planter box increases by 1, so the total number of flowers after n weeks is 2 + n. Wait, that would be if she adds 1 flower each week. But the problem says "plants one more flower", which could mean that each week, she plants one additional flower compared to the previous week. Wait, the wording is ambiguous, but let's re-read: "Jackie plants 2 flowers in a planter box. Every week, she plants one more flower in the planter box. How many flowers does Jackie have in her planter box after 5 weeks?"

Ah, maybe "plants one more flower" means that each week, the number of flowers she plants is 2, then 3, then 4, etc., and the total number of flowers is the sum of these. Wait, no, maybe "has in her planter box" is the total number, so it's an arithmetic sequence where the first term is 2, common difference 1, and we need the term after 5 weeks. Wait, if after 0 weeks (initial), it's 2; after 1 week, 3; after 2 weeks, 4; ... after 5 weeks, it's 2 + 5 = 7? Wait, that would be if each week, the total number increases by 1. But that seems too simple. Wait, let's check with the sum. If she plants 2 initially, then each week adds 1, so after 5 weeks, the total is 2 + 3 + 4 + 5 + 6 + 7? No, that's 6 terms. Wait, maybe the problem is that "after 5 weeks" means 5 weeks of planting, so initial (week 0) plus 5 weeks, total 6 plantings. But the sum of 2 + 3 + 4 + 5 + 6 + 7 is (2 + 7) * 6 / 2 = 9 * 3 = 27. But that seems high. Wait, maybe the problem is that "every week, she plants one more flower" means that each week, she plants 1 additional flower, so the number of flowers planted each week is 2, 3, 4, 5, 6 for 5 weeks (week 1 to week 5). Then the total is 2 + 3 + 4 + 5 + 6 = (2 + 6) * 5 / 2 = 8 * 5 / 2 = 20. But that's also high. Wait, maybe the problem is that "has in her planter box" is the total number, which is an arithmetic sequence with first term 2, common difference 1, and we need the 6th term (after 5 weeks, since initial is week 0). The nth term of an arithmetic sequence is \( a_n = a_1 + (n - 1)d \). So \( a_6 = 2 + (6 - 1)*1 = 7 \). So after 5 weeks (week 5), the total is 7. That seems more reasonable. So the answer is 7.

Wait, let's verify:

- Start: 2 flowers (week 0)

- Week 1: 2 + 1 = 3

- Week 2: 3 + 1 = 4

- Week 3: 4 + 1 = 5

- Week 4: 5 + 1 = 6

- Week 5: 6 + 1 = 7

Yes, that makes sense. So after 5 weeks, she has 7 flowers.

Step2: Calculate the total after 5 weeks

Using the arithmetic sequence formula for the nth term, where \( a_1 = 2 \) (initial), \( d = 1 \) (common difference), and we need the term at week 5 (which is the 6th term, but wait, week 0 is term 1, week 1 is term 2, ..., week 5 is term 6? No, maybe week 0 is term 0, so term n (after n weeks) is \( a_n = 2 + n \). So after 5 weeks, \( a_5 = 2 + 5 = 7 \).

Answer:

7