John has taken out a loan for college. He started paying off the loan with a first payment of $100. Each month he pays, he wants to pay back 1.1 times as the amount he paid the month before. Explain to John how to represent his first 20 payments in sigma notation. Then explain how to find the sum of his first 20 payments, using complete sentences. Explain why this series is convergent or divergent.

Question

amistre64 · Answer

any ideas?

anonymous · Answer

to multiply 100 and 1.1? i don't get what to do with sigma notation in this for 20 payments..

amistre64 · Answer

well, if we can produce a rule, all sigma tell us to do is to add them up.

amistre64 · Answer

an = an-1 (1.1)  is an implicit rule,  we need to make it explicit.

anonymous · Answer

how?

amistre64 · Answer

well, the simplest way is to see if we can spot a pattern .... other ways involve understanding what type of sequence is generated ... etc.  what does your material present you with?

amistre64 · Answer

$$a_0=100(1.1)^0\
a_1=100(1.1)^1\
a_2=100(1.1)^2\
a_3=100(1.1)^3\
a_4=100(1.1)^4\
...\
a_n=100(1.1)^n
$$
does this seem like a reasonable pattern?

anonymous · Answer

yes, am I supposed to add these?

amistre64 · Answer

ideally, yes.  but you are expected to do so with your intricate knowledge of summation notation and convergence properties.