Suppose you invest 100.00 in a bank that pays you 5 %...

Suppose you invest $\$ 100.00$ in a bank that pays you $5 \%$ interest compounded annually. The balance in the account after $k$ years is given by $a_{k}=100(1+0.05)^{k} .$ To the nearest cent, determine the first five terms of the sequence, starting at $k=0 .$ What does $k=0$ mean in practical terms? Determine whether the sequence is bounded. Determine whether the sequence is increasing, decreasing, or not monotonic.

Key Concepts

Compound Interest

Compound interest is the process where interest is added to the principal of a deposit or loan so that, from that moment on, the interest that has been added also earns interest. This results in exponential growth because the accumulation of interest accelerates as both the original amount and the accumulated interest earn more interest over time.

Exponential Growth

Exponential growth describes a process where a quantity increases by a fixed percentage over regular time intervals, leading to a rapid increase as time goes on. In the context of investments and compound interest, each term in the sequence is a multiple of the previous term, reflecting the compounded nature of the growth.

Initial Value in Sequences

The initial value in sequences, often represented by an index such as k = 0, serves as the starting point or baseline from which subsequent values are calculated. In financial contexts, this initial term typically represents the original investment or deposit before any interest has been accrued.

Bounded Sequence

A bounded sequence is one where all its terms lie within a fixed interval, meaning there exists a real number that serves as an upper limit and possibly a lower limit as well. Determining the boundedness of a sequence helps in understanding its long-term behavior, particularly whether it grows without limit or remains constrained.

Monotonic Sequence

A monotonic sequence is one that is either entirely non-increasing or non-decreasing throughout its domain. In the context of financial growth models like compound interest, the sequence is generally increasing, as each successive term is greater than the previous term due to the accumulation of interest.

Transcript

00:01 Here in this question the given part is ak where ak is equal to 100 multiplied by 1 plus 0 .05 rastey par k and by using this we have to evaluate first five terms by taking k as k as now we also have to tell what is the meaning of k is equal to 0 in practical terms and as well as we also have to determine whether the sequence ak is bounded or not and also so, find out the monotonicity of the sequence.

01:08 That is, the given sequence is whether increasing, decreasing or not monotonic.

01:16 So we will start with the given part that is ak is equal to 1001 plus 0 .05k.

01:25 So to evaluate first five terms, let's put the value of k is equals to 0, 1, 2, 3, 4 respectively.

01:36 Now when we put k is equal to 0, we will find a not as the first term and this is equal to 1001 plus 0 .05 raised the power 0.

01:51 So this is equals to 100.

01:53 Now when we put k is equal to 1 we'll get a1 as 100 multiplied by 1 plus 0 .05 rasty power 1 and which is equals to 105.

02:06 Now, a2 will be 100 multiplied by 1 plus 0 .05 rastert power 2 and this value will be equal to 11 .25 approximately.

02:22 Now when we put k is equal to 3 we'll get a3 as 100 multiplied by 1 plus 0 .25 0 .05 raised the power 3 and this value will approximately equals to 115 .76.

02:38 And as we put k is equal to 4, we will get a4 as 100 multiplied by 1 plus 0 .05 raised the power 4 and this value will approximately equal to 1 to 1 .15.

02:56 So these are our required...

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