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Math  /  Algebra

Question12. Lisa vill spara till en ny cykel som kostar 10000 kronor. Hon öppnar ett sparkonto och sätter in samma belopp CC kronor i början av varje år under 5 år. Kontot har en årlig ränta på 3%3 \%.
Frågor:
1. Hur mycket är den första insättningen värd efter 5 år, om räntan är 3%3 \% per år?
2. Hur mycket är den andra insättningen värd efter 4 år?
3. Hur kan vi skriva en formel för det totala värdet av Lisas insättningar efter 5 år?
4. Om Lisa vill spara exakt 10000 kronor, vilket årligt belopp CC behöver hon sätta in?
Lös uppgiften med en geometrisk summa.

Studdy Solution

STEP 1

What is this asking? Lisa wants to save \$10,000 for a bike by depositing the same amount of money, \(\$C\), every year for 5 years, with a 3% annual interest rate.
How much does she need to deposit each year? Watch out! Each deposit earns interest for a *different* number of years!

STEP 2

1. Calculate the value of the first deposit after 5 years.
2. Calculate the value of the second deposit after 4 years.
3. Derive the formula for the total value after 5 years.
4. Calculate the required annual deposit.

STEP 3

Alright, let's **start** with Lisa's **first deposit**, $C\$C.
It sits in the bank for a full **5 years**, growing at a **3% annual interest rate**.

STEP 4

Remember, a **3% increase** means multiplying by 1+0.03=1.031 + 0.03 = 1.03.

STEP 5

After **one year**, the first deposit grows to C⋅1.03C \cdot 1.03.
After **two years**, it's C⋅1.03⋅1.03=C⋅(1.03)2C \cdot 1.03 \cdot 1.03 = C \cdot (1.03)^2.
See the pattern?

STEP 6

So, after **five years**, the **first deposit** will be worth C⋅(1.03)5C \cdot (1.03)^5.

STEP 7

Now, onto the **second deposit**, also $C\$C.
This one stays in the bank for **4 years**.

STEP 8

Using the same logic as before, after **four years**, the **second deposit** grows to C⋅(1.03)4C \cdot (1.03)^4.

STEP 9

The **third deposit** earns interest for **3 years**, becoming C⋅(1.03)3C \cdot (1.03)^3.
The **fourth deposit** earns interest for **2 years**, becoming C⋅(1.03)2C \cdot (1.03)^2.
And the **fifth deposit** earns interest for just **1 year**, becoming C⋅1.03C \cdot 1.03.

STEP 10

To find the **total value**, we **add up** all the future values of the deposits: C⋅(1.03)5+C⋅(1.03)4+C⋅(1.03)3+C⋅(1.03)2+C⋅1.03 C \cdot (1.03)^5 + C \cdot (1.03)^4 + C \cdot (1.03)^3 + C \cdot (1.03)^2 + C \cdot 1.03

STEP 11

We can **factor out** CC to get: C⋅[(1.03)5+(1.03)4+(1.03)3+(1.03)2+1.03] C \cdot \left[ (1.03)^5 + (1.03)^4 + (1.03)^3 + (1.03)^2 + 1.03 \right]

STEP 12

This is a **geometric series**!
We can use the formula for the sum of a geometric series: Sn=a⋅1−rn1−r S_n = a \cdot \frac{1 - r^n}{1 - r} where aa is the **first term**, rr is the **common ratio**, and nn is the **number of terms**.

STEP 13

In our case, a=1.03a = 1.03, r=1.03r = 1.03, and n=5n = 5.
So, the **sum** inside the brackets is: 1.03⋅1−(1.03)51−1.03=1.03⋅1−1.15927−0.03≈1.03⋅−0.15927−0.03≈1.03⋅5.309=5.46827 1.03 \cdot \frac{1 - (1.03)^5}{1 - 1.03} = 1.03 \cdot \frac{1 - 1.15927}{ -0.03} \approx 1.03 \cdot \frac{-0.15927}{-0.03} \approx 1.03 \cdot 5.309 = 5.46827

STEP 14

Therefore, the **total value** is approximately C⋅5.46827C \cdot 5.46827.

STEP 15

Lisa wants this **total value** to be $10,000\$10,000.
So, we set up the equation: C⋅5.46827=10000 C \cdot 5.46827 = 10000

STEP 16

To find CC, we **divide both sides** by 5.468275.46827: C=100005.46827≈1828.75 C = \frac{10000}{5.46827} \approx 1828.75

STEP 17

Lisa needs to deposit approximately $1828.75\$1828.75 each year.

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