Mixing single and periodic payments

I start an interest-bearing savings account with $5,000 of capital. Then I make regular payments of $500 at the end of each quarter.

What will be the balance after 10 years if interest is at 6 % compounded annually?

Step 1: Calculate the future value of the initial balance after 10 years.

Answer: $8,954.24.

Step 2: Calculate the future value of the periodic payments.

Answer: $26,947.53.

Conclusion

There will be $8,954.24 + $26,947.53 = $35,901.77 in the acount after 10 years.

What will be the balance after 10 years if each payment will increase by $10 resulting in the series $500, $510, $520, ... ?

Step 1: Calculate the future value of the initial balance after 10 years as in the previous example.

Step 2: Calculate the future value of the periodic payments.

Answer: $36,416.79.

Conclusion

There will be $8,954.24 + $36,416.79 = $45,371.03 in the account after 10 years if periodic payments increase by $10 each quarter.

Given the previous data I also expect to make an extra deposit of $2,000 at the end of the 7th year. How do I add its future value?

Step 1: Calculate the future value for the initial amount and the series of periodic payments as in the previous example.

Step 2: Calculate the future value of the extra $2,000 after 10 - 7 = 3 years.

Conclusion

After 10 years there will be $8,954.24 + $36,416.79 + $2,382.03 = $47,753.06 in the account.