Let's solve this problem step-by-step.

Given:

• Savings increase weekly in an arithmetic sequence.
• Week 1 savings: 5p, Week 2 savings: 7p, Week 3 savings: 9p, etc.
• The common difference (d) can be calculated as the difference between consecutive weeks' savings.

Part (a): Find the amount saved in Week 200

The general formula for the $n$-th term of an arithmetic sequence is:

$a_n = a_1 + (n-1) \cdot d$

where:

• $a_1 = 5p$ (savings in Week 1)
• $d = 7p - 5p = 2p$ (common difference)
• $n = 200$ (for Week 200)

Substitute the values:

$a_{200} = 5 + (200 - 1) \cdot 2$

Let's calculate it.

The amount saved in Week 200 is 403p.

Part (b): Calculate the total savings over the complete 200-week period

The formula for the sum of the first $n$ terms of an arithmetic sequence is:

$S_n = \frac{n}{2} \times (a_1 + a_n)$

where:

• $n = 200$
• $a_1 = 5p$
• $a_{200} = 403p$ (from part (a))

Substitute the values to calculate the total savings:

$S_{200} = \frac{200}{2} \times (5 + 403)$

Let's calculate this value.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, I can guide you on how to calculate the total savings manually.

Manual Calculation for Part (b)

The formula for the sum of the first $n$ terms of an arithmetic sequence is:

$S_n = \frac{n}{2} \times (a_1 + a_n)$

where:

• $n = 200$
• $a_1 = 5p$
• $a_{200} = 403p$

Substitute the values:

$S_{200} = \frac{200}{2} \times (5 + 403) = 100 \times 408 = 40800p$

So, the total savings over the 200-week period is 40800p, which is equivalent to £408.00.

Would you like any more details or have any other questions?

Related Questions:

1. What is the formula for an arithmetic sequence?
2. How do you calculate the common difference in an arithmetic sequence?
3. How can you find the sum of an arithmetic series for any number of terms?
4. What are other types of sequences besides arithmetic ones?
5. Can this arithmetic sequence formula be applied to real-life savings or investment plans?

Tip:

Understanding arithmetic sequences helps in calculating sums and predicting values in various practical scenarios, such as finance, planning, and resource allocation.