Compound Interest Recurrence Relations
Emily invests $1000 in a savings account that earns 5% interest compounded annually. Write the recurrence relation that describes the amount of money in Emily's account at the end of each year.
t0= $1000, tn+1=tn x 1.05
t0=$1,000
t1=$1050
t2=$1,102.50
t3=$1,157.63
Amy invests $1000 in a savings account with 5% interest compounded annually. Write the nth term rule for the amount in the account after nnn years. How much will be in the account after 3 years?
tn=1000 x 1.05n
t3=1000 x 1.053
= $1157.63
Liam invests $1000 in a bank account that earns 4% simple interest annually. Write the recurrence relation for the account balance and find the amount in the account after 1, 2, and 3 years.
t0= $1000, tn+1=tn +40
t0=$1,000
t1=$1040
t2=$1,080
t3=$1,120
Liam invests $1000 in a savings account with 4% simple interest annually. Write the nth term rule for the amount in the account after nnn years. How much will be in the account after 5 years?
tn=1000 + 40n
t5=1000 + 40 x 5
= $1200
Option A: Liam invests $1000 at 4% simple interest per annum.
Option B: He also has the option to invest the same amount at 4% compound interest per annum (compounded annually).
Which option gives him more money after 3 years, and how much more?
Option A (Simple):
t3=1000 + 40 x 3 = $1,120
Option B (Compound):
t3= 1000 x 1.043 = $1,124.86
Difference:
1124.86 - 1120 = 4.86
Compound interest earns $4.86 more
Olivia deposits $2000 in a bank account with an interest rate of 7% compounded annually. Write the recurrence relation for the amount of money in Olivia's account after each year.
t0= $2000, tn+1=tn x 1.07
t0=$2,000
t1=$2,140
t2=$2,289.80
t3=$2450.09
Oliver deposits $1500 in a bank account with an interest rate of 6% compounded annually. Write the nth term rule for the balance after nnn years. How much will be in the account after 5 years?
tn=1500 x 1.06n
t5=1500 x 1.065
= $2007.35
Olivia deposits $2500 into an account that earns 5% simple interest annually. Write the recurrence relation for the balance and find the amount after 1, 2, and 3 years.
t0= $2500, tn+1=tn +125
t0=$2,500
t1=$2,625
t2=$2,750
t3=$2,875
Olivia deposits $2500 into an account that earns 5% simple interest annually. Write the nth term rule for the balance after nnn years. How much will be in the account after 8 years?
tn=2500 + 125n
t8=2500 + 125 x 8
= $3,500
Option A: Olivia invests $1500 at 5% simple interest per annum.
Option B: She has the option to invest the same amount at 4% compound interest per annum (compounded annually).
Which plan gives her more money after 5 years, and how much more?
Option A (Simple):
t5=1500 + 75 x 5 = $1,875
Option B (Compound):
t5= 1500 x 1.045 = $1,824.46
Difference:
1875 - 1824.46 = 50.54
Simple interest earns $50.54 more
Sophie invests $500 into an account with a 4% interest rate compounded annually. Write the recurrence relation for the amount in her account at the end of each year.
t0= $500, tn+1=tn x 1.04
t0=$500
t1=$520
t2=$540.80
t3=$562.43
Evelyn invests $800 into an account that earns 4% interest compounded annually. Write the nth term rule for the amount in the account after nnn years. How much will be in the account after 7 years?
tn=800 x 1.04n
t7=800 x 1.047
= $1052.29
Emma invests $1500 in a savings account with an interest rate of 6% simple interest annually. Write the recurrence relation and find the balance after 1, 2, and 3 years.
t0= $1500, tn+1=tn +90
t0=$1,500
t1=$1,590
t2=$1,680
t3=$1,770
Emma invests $1500 in a savings account with 6% simple interest annually. Write the nth term rule and find the balance after 10 years.
tn=1500 + 90n
t10=1500 + 90 x 10
= $2400
Option A: Emma invests $2000 at 3.5% simple interest per annum.
Option B: She has the option to invest the same amount at 3% compound interest per annum (compounded annually).
Which option gives her more money after 6 years, and by how much?
Option A (Simple):
t6=2000 + 70 x 6 = $2,420
Option B (Compound):
t6= 2000 x 1.036 = $2,385.48
Difference:
2420- 2385.48 = 34.52
Simple interest earns $34.52 more
David invests $5000 in a bank account with an interest rate of 8.5% compounded annually. Write the recurrence relation for the balance at the end of each year.
t0= $5000, tn+1=tn x 1.085
t0=$5,000
t1=$5,400
t2=$5,832
t3=$6,298.56
Lucas deposits $5000 into a bank account that earns 7.7% interest compounded annually. Write the nth term rule for the balance after nnn years. How much will be in the account after 10 years?
tn=5000 x 1.077n
t10=5000 x 1.07710
= $10,498.49
David invests $4000 into an account that earns 3.6% simple interest annually. Write the recurrence relation for the balance and find the amount in the account after 1, 2, and 3 years.
t0= $4,000, tn+1=tn + 144
t0=$4,000
t1=$4,144
t2=$4,288
t3=$4,432
David invests $4000 into a savings account that earns 3% simple interest annually. Write the nth term rule for the balance after nnn years. How much will be in the account after 7 years?
tn=4000 + 120n
t7=4000 + 120 x 7
= $4840
Option A: David invests $5000 at 4.75% simple interest per annum.
Option B: He can also invest the same amount at 4% compound interest per annum (compounded annually).
Which plan will give him more after 8 years, and how much more?
Option A (Simple):
t8=5000 + 237.5 x 8 = $6,900
Option B (Compound):
t8= 5000 x 1.048 = $6,843.28
Difference:
6900 - 6843.28 = 56.72
Simple interest earns $56.72 more
Sarah deposits $10,000 into an account that earns 4.6% interest compounded annually. Write the recurrence relation for the amount in her account after each year.
t0= $10,000, tn+1=tn x 1.046
t0=$10,000
t1=$10,460
t2=$10,941.16
t3=$11,444.45
Emma invests $10,000 into an account with 8.2% interest compounded annually. Write the nth term rule for the amount after n years. How much will be in the account after 6 years? How much interest is earned?
tn=10000 x 1.082n
t6=10000 x 1.0826
= $16,045.88
Interest = 16,045.88 - 10,000
= $6,045.88
Sophia deposits $8000 into a savings account that earns 7.2% simple interest annually. Write the recurrence relation and find the balance after 1, 2, and 3 years.
t0= $8,000, tn+1=tn + 576
t0= $8,000
t1=$8,576
t2=$9,152
t3=$9728
Sophia deposits $8000 into a savings account that earns 7.4% simple interest annually. Write the nth term rule, find the balance after 12 years and how much interest was earned.
tn=8000 + 592n
t12=8000 + 592 x 12
= $15,104
Interest = $7,104
Option A: Sophia invests $10,000 at 3.75% simple interest per annum.
Option B: She can also invest the same amount at 3.5% compound interest per annum (compounded annually).
Which plan gives her more money after 10 years, and by how much?
Option A (Simple):
t10=10,000 + 375 x 10 = $13,750
Option B (Compound):
t10= 10,000 x 1.03510 = $14,106.08
Difference:
14106.08 - 13750 = 356.08
Compound interest earns $356.08 more