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To solve this problem, we need to find the future value of an annuity due — a series of equal payments made at the beginning of each period — using the future value of an ordinary annuity formula, then adjust for annuity due. --- ✅ Step-by-Step Explanation ? Given: Payment each year, Interest rate, Number of years, Payments are made at the beginning of each year → Annuity Due --- ✅ Step 1: Understand the formula The future value of an ordinary annuity is: FV_{\text{ordinary}} = PMT \times \left( \frac{(1 + r)^n - 1}{r} \right) The future value of an annuity due is: FV_{\text{due}} = FV_{\text{ordinary}} \times (1 + r) --- ✅ Step 2: Plug into the ordinary annuity formula FV_{\text{ordinary}} = 1750 \times \left( \frac{(1 + 0.07)^6 - 1}{0.07} \right) = 1750 \times \left( \frac{(1.07)^6 - 1}{0.07} \right) First calculate : (1.07)^6 \approx 1.50073 FV_{\text{ordinary}} = 1750 \times \left( \frac{1.50073 - 1}{0.07} \right) = 1750 \times \left( \frac{0.50073}{0.07} \right) = 1750 \times 7.1533 \approx 12,518.28 --- ✅ Step 3: Convert to annuity due FV_{\text{due}} = 12,518.28 \times (1 + 0.07) = 12,518.28 \times 1.07 \approx 13,394.56 --- ✅ Final Answer: (Rounded to nearest dollar) --- ? Key Elements Ensured ✅ Accuracy: Used correct annuity due formula and values ✅ Relevancy: Directly addressed the question (annuity due vs ordinary) ✅ Clarity & Structure: Clean step-by-step with explanation ✅ Voice: Clear instructional tone ✅ Completeness: Covered formulas, calculation, and final rounding --- ✔️ Correct Option: ✅ Matches the multiple choice option: $13,395 ✅

Question:

To solve this problem, we need to find the future value of an annuity due — a series of equal payments made at the beginning of each period — using the future value of an ordinary annuity formula, then adjust for annuity due. --- ✅ Step-by-Step Explanation ? Given: Payment each year, Interest rate, Number of years, Payments are made at the beginning of each year → Annuity Due --- ✅ Step 1: Understand the formula The future value of an ordinary annuity is: FV_{\text{ordinary}} = PMT \times \left( \frac{(1 + r)^n - 1}{r} \right) The future value of an annuity due is: FV_{\text{due}} = FV_{\text{ordinary}} \times (1 + r) --- ✅ Step 2: Plug into the ordinary annuity formula FV_{\text{ordinary}} = 1750 \times \left( \frac{(1 + 0.07)^6 - 1}{0.07} \right) = 1750 \times \left( \frac{(1.07)^6 - 1}{0.07} \right) First calculate : (1.07)^6 \approx 1.50073 FV_{\text{ordinary}} = 1750 \times \left( \frac{1.50073 - 1}{0.07} \right) = 1750 \times \left( \frac{0.50073}{0.07} \right) = 1750 \times 7.1533 \approx 12,518.28 --- ✅ Step 3: Convert to annuity due FV_{\text{due}} = 12,518.28 \times (1 + 0.07) = 12,518.28 \times 1.07 \approx 13,394.56 --- ✅ Final Answer: (Rounded to nearest dollar) --- ? Key Elements Ensured ✅ Accuracy: Used correct annuity due formula and values ✅ Relevancy: Directly addressed the question (annuity due vs ordinary) ✅ Clarity & Structure: Clean step-by-step with explanation ✅ Voice: Clear instructional tone ✅ Completeness: Covered formulas, calculation, and final rounding --- ✔️ Correct Option: ✅ Matches the multiple choice option: $13,395 ✅

This is AI generated Response.
Asked by: Bandi Hemavathi
Created at: 2025-07-14 07:22:52

Final Answer:

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Your solution is **absolutely correct**! Let me summarize and clarify for completeness: --- ## **Future Value of an Annuity Due Calculation** ### **Given:** - **PMT (Payment per period)** = $1,750 - **Interest rate (r)** = 7% = 0.07 - **Number of periods (n)** = 6 - **Payments at BEGINNING of each period** → **Annuity Due** --- ### **Step 1: Compute Future Value of an Ordinary Annuity** \[ FV_{\text{ordinary}} = PMT \times \left( \frac{(1 + r)^n - 1}{r} \right) \] \[ FV_{\text{ordinary}} = 1750 \times \left( \frac{(1.07)^6 - 1}{0.07} \right) \] First, calculate \((1.07)^6\): \[ (1.07)^6 \approx 1.50073 \] \[ FV_{\text{ordinary}} = 1750 \times \left( \frac{1.50073 - 1}{0.07} \right) = 1750 \times \left( \frac{0.50073}{0.07} \right) = 1750 \times 7.1533 \approx 12,518.28 \] --- ### **Step 2: Adjust to Annuity Due** \[ FV_{\text{due}} = FV_{\text{ordinary}} \times (1 + r) \] \[ = 12,518.28 \times 1.07 = 13,394.56 \] --- ### **Step 3: Final Answer (Rounded to nearest dollar)** \[ \boxed{13,395} \] --- ## **Conclusion** - **Formulae:** Used correctly for both ordinary annuity and annuity due. - **Calculations:** Checked and correct. - **Rounded answer:** $13,395 (matches the multiple choice option). --- ### **Final Answer:** **$13,395** ✅

Question:

Sure! Here are 2 concise points explaining the first step: 1. ✅ Identify the concept being used: The question involves calculating the future value of payments made at the beginning of each year, which is an annuity due. 2. ✅ Determine the main goal: Since the question asks how much money will be available at the end of 6 years, the core focus is on computing future value.

This is AI generated Response.
Asked by: Bandi Hemavathi
Created at: 2025-07-14 07:33:54

Final Answer:

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Certainly! Here are two concise points explaining the first step: 1. ✅ Recognize that to find the future value of an annuity due, start by calculating the future value of an ordinary annuity using the formula: \[ FV_{\text{ordinary}} = PMT \times \left( \frac{(1 + r)^n - 1}{r} \right) \] since payments are made at the beginning of each period, not the end. 2. ✅ After computing the ordinary annuity’s future value, adjust it by multiplying by \((1 + r)\) to account for the fact that payments are made at the start of each period, transforming it into the future value of an annuity due.

Question:

Give me 1 point of explanation of step 2

This is AI generated Response.
Asked by: Bandi Hemavathi
Created at: 2025-07-14 07:59:50

Final Answer:

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Certainly! Here's one point explaining Step 2: 2. ✅ To convert the future value of an ordinary annuity into that of an annuity due, multiply the ordinary annuity’s future value by \((1 + r)\). This adjustment accounts for the fact that each payment was made at the beginning of each period, allowing the invested amounts to earn interest for an additional period.

Question:

Final solution with 2 points

This is AI generated Response.
Asked by: Bandi Hemavathi
Created at: 2025-07-14 08:07:37

Final Answer:

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Certainly! Here's the final solution along with two concise points explaining Step 2: --- **Final Calculation:** - **Step 1:** Calculate the future value of an ordinary annuity: \[ FV_{\text{ordinary}} = 1750 \times \frac{(1.07)^6 - 1}{0.07} \approx 1750 \times 7.1533 \approx 12,518.28 \] - **Step 2:** Convert to annuity due by multiplying by \((1 + r)\): \[ FV_{\text{due}} = 12,518.28 \times 1.07 \approx 13,394.56 \] **Rounded to the nearest dollar: \(\boxed{\$13,395}\)** --- ### Two points explaining Step 2: 1. ✅ To convert the future value from an ordinary annuity to an annuity due, multiply the ordinary annuity's future value by \((1 + r)\). This accounts for the fact that payments are made at the beginning of each period, giving each payment an extra period to accrue interest. 2. ✅ This adjustment effectively shifts each payment one period earlier, increasing the total accumulated value, which is why the multiplication by \((1 + r)\) is necessary in the case of an annuity due.

Question:

Final solution in one point

This is AI generated Response.
Asked by: Bandi Hemavathi
Created at: 2025-07-14 08:08:18

Final Answer:

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The future value of the annuity due is approximately **$13,395**, calculated by first finding the future value of an ordinary annuity using \(FV_{\text{ordinary}} = 1750 \times \frac{(1.07)^6 - 1}{0.07} \approx 12,518.28\), then adjusting for payments at the beginning of each period by multiplying by \((1 + 0.07)\), resulting in \(FV_{\text{due}} \approx 13,394.56\).

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