The Amount of Money in the Account Every Year When ₹10,000 is Deposited at 8% Compound Interest per Annum
Step-by-Step NCERT Solution Using the Compound Interest Formula
Key Takeaways
- Learn how to calculate the amount using the Compound Interest formula.
- Find the amount after the 1st, 2nd, and 3rd year.
- Understand whether the obtained sequence forms an Arithmetic Progression (AP).
- Step-by-step NCERT solution with detailed calculations.
- Useful for CBSE, ICSE, State Boards, and competitive examinations.
Question
Find the amount of money in the account every year when ₹10,000 is deposited at a compound interest rate of 8% per annum. Also determine whether the yearly amounts form an Arithmetic Progression (AP).
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| Math exercise: compound interest question |
Introduction
Compound interest is one of the most important concepts in commercial mathematics. Unlike simple interest, compound interest is calculated not only on the original principal but also on the accumulated interest from previous years. As a result, the total amount grows faster every year.
In this problem, a principal amount of ₹10,000 is invested at an annual compound interest rate of 8%. We will calculate the amount after each year and determine whether the resulting sequence is an Arithmetic Progression (AP).
Given Data
Principal (P)
₹10,000
Rate (R)
8% per annum
Time (T)
1 Year, 2 Years, and 3 Years
Interest Type
Compound Interest
Compound Interest Formula
The amount after T years at compound interest is calculated using the following formula:
A = P(1 + R/100)T
Where:
- A = Amount after T years
- P = Principal amount
- R = Rate of interest (per annum)
- T = Time in years
Values to Substitute
- P = ₹10,000
- R = 8%
- T = 1, 2, and 3 years
Now, let us calculate the amount for each year one by one using the formula above.
Step 1: Calculate the Amount After 1 Year
Using the compound interest formula,
A = P(1 + R/100)T
Substitute the given values:
P = ₹10,000
R = 8%
T = 1 year
Now,
A = 10000(1 + 8/100)1
= 10000(108/100)
= 10000 × 1.08
= ₹10,800
Principal
₹10,000
Interest Rate
8% per annum
Time
1 Year
Amount
₹10,800
Result After 1 Year
Final Answer
A₁ = ₹10,800
Therefore, the amount in the account after the first year is ₹10,800.
Calculation Summary
| Quantity | Value |
|---|---|
| Principal (P) | ₹10,000 |
| Rate (R) | 8% per annum |
| Time (T) | 1 Year |
| Formula Used | A = P(1 + R/100)T |
| Amount After 1 Year | ₹10,800 |
Observation
After one year, the investment grows from ₹10,000 to ₹10,800 because the interest earned is added to the principal. In the next year, interest will be calculated on ₹10,800 instead of ₹10,000, which is why this is called compound interest.
Step 2: Calculate the Amount After 2 Years
Now calculate the amount after 2 years using the Compound Interest formula.
A = P(1 + R/100)T
Substitute the given values:
P = ₹10,000
R = 8%
T = 2 years
Now simplify the expression step by step.
A = 10000(1 + 8/100)2
= 10000(108/100)2
= 10000 × (1.08)2
= 10000 × 1.1664
= ₹11,664
Principal
₹10,000
Interest Rate
8% per annum
Time
2 Years
Amount
₹11,664
Step 3: Calculate the Amount After 3 Years
Next, calculate the amount after 3 years.
A = P(1 + R/100)T
Substitute the values:
P = ₹10,000
R = 8%
T = 3 years
A = 10000(1 + 8/100)3
= 10000(108/100)3
= 10000 × (1.08)3
= 10000 × 1.259712
= ₹12,597.12
Principal
₹10,000
Interest Rate
8% per annum
Time
3 Years
Amount
₹12,597.12
Amounts After Each Year
The following table summarizes the amount in the account at the end of each year.
| Year | Formula | Amount (₹) |
|---|---|---|
| 1st Year | 10000 × (1.08)1 | 10,800 |
| 2nd Year | 10000 × (1.08)2 | 11,664 |
| 3rd Year | 10000 × (1.08)3 | 12,597.12 |
Observation
Notice that the amount increases every year because interest is earned not only on the original principal but also on the interest accumulated in previous years. This is the key feature of compound interest.
Step 4: Check Whether the Given Sequence Forms an Arithmetic Progression (AP)
The amounts obtained after each year are:
₹10,800, ₹11,664, ₹12,597.12
To determine whether these amounts form an Arithmetic Progression (AP), we calculate the difference between consecutive terms.
Formula
Common Difference (d) = an+1 − an
Difference Between Consecutive Terms
a₁ = ₹10,800
a₂ = ₹11,664
a₃ = ₹12,597.12
Now calculate the differences.
a₂ − a₁
= 11,664 − 10,800
= 864
a₃ − a₂
= 12,597.12 − 11,664
= 933.12
Comparison of Differences
| Difference | Value |
|---|---|
| a₂ − a₁ | 864 |
| a₃ − a₂ | 933.12 |
864 ≠ 933.12
Since the consecutive differences are not equal, the sequence does not satisfy the definition of an Arithmetic Progression.
Mathematical Conclusion
Final Result
a₂ − a₁ ≠ a₃ − a₂
Therefore,
The sequence is NOT an Arithmetic Progression (AP).
Important Concept
Arithmetic Progression
An AP has a constant common difference between consecutive terms.
Compound Interest
Interest is calculated on both the principal and the accumulated interest.
Growth Pattern
Compound interest produces exponential growth rather than linear growth.
Result
Hence, the yearly amounts cannot form an AP.
Memory Tips
- ✔ Simple Interest generally increases by the same amount every year.
- ✔ Compound Interest increases by larger amounts each year.
- ✔ If the common difference changes, the sequence is not an AP.
- ✔ Always calculate at least two consecutive differences before concluding.
Exam Tips
- Write the compound interest formula before starting the solution.
- Calculate each year's amount carefully.
- Use the exact values while checking the common difference.
- State the mathematical reason for your conclusion.
- Remember: Compound interest usually produces a Geometric Progression (GP), not an Arithmetic Progression (AP).
Frequently Asked Questions (FAQs)
1. What is Compound Interest?
Compound Interest (CI) is the interest calculated on both the original principal and the accumulated interest from previous years. Therefore, the amount increases faster than in simple interest.
2. What is the formula for Compound Interest?
The amount after T years is calculated using the formula:
A = P(1 + R/100)T
Where:
- A = Amount
- P = Principal
- R = Rate of Interest
- T = Time (Years)
3. What is the amount after 1 year?
The amount after one year is: ₹10,800
4. What is the amount after 2 years?
The amount after two years is: ₹11,664
5. What is the amount after 3 years?
The amount after three years is: ₹12,597.12
6. Does the sequence form an Arithmetic Progression?
No. Since the common differences are not equal, the yearly amounts do not form an Arithmetic Progression (AP).
7. Why doesn't Compound Interest form an AP?
Because each year's interest is calculated on the previous year's amount, the increase is not constant. The sequence follows exponential growth rather than linear growth.
8. Which chapter does this question belong to?
This question belongs to the chapter on Arithmetic Progressions, where students determine whether a given sequence forms an AP.
Quick Revision
- ✔ Principal (P) = ₹10,000
- ✔ Rate of Interest (R) = 8% per annum
- ✔ Formula: A = P(1 + R/100)T
- ✔ Amount after 1 Year = ₹10,800
- ✔ Amount after 2 Years = ₹11,664
- ✔ Amount after 3 Years = ₹12,597.12
- ✔ The sequence does not form an Arithmetic Progression.
Summary Table
| Particular | Value |
|---|---|
| Principal | ₹10,000 |
| Rate of Interest | 8% per annum |
| Compound Interest Formula | A = P(1 + R/100)T |
| Amount After 1 Year | ₹10,800 |
| Amount After 2 Years | ₹11,664 |
| Amount After 3 Years | ₹12,597.12 |
| Common Difference | Not Constant |
| Final Conclusion | Not an Arithmetic Progression (AP) |
Conclusion
Using the Compound Interest formula, we found that the amount in the account increases from ₹10,000 to ₹10,800 after the first year, ₹11,664 after the second year, and ₹12,597.12 after the third year. Since compound interest is calculated on both the principal and the accumulated interest, the yearly increase is not constant.
When we compare the consecutive differences, we observe that they are unequal. Therefore, the sequence of yearly amounts does not satisfy the definition of an Arithmetic Progression. Instead, it represents an example of exponential growth produced by compound interest.
Final Answer
Amounts:
₹10,800, ₹11,664, ₹12,597.12
These amounts do NOT form an Arithmetic Progression (AP).
References
- NCERT Mathematics – Class 10
- CBSE Mathematics Curriculum
- NCERT Exemplar Problems – Arithmetic Progressions
- Commercial Mathematics – Compound Interest
- Standard School Mathematics Textbooks