A. Show that the first four terms of the sequence defined by...

a. Show that the first four terms of the sequence defined by $a_{n}=n^{2}$ are $$1,4,9,16, \dots$$ b. Show that the first four terms of the sequence defined by $a_{n}=n^{2}+(n-1)(n-2)(n-3)(n-4)$ are also $$1,4,9,16, \dots$$ c. Find a sequence whose first six terms are the same as those of $a_{n}=n^{2}$ but whose succeeding terms differ from this sequence. d. Find two different sequences that begin $$2,4,8,16, \dots$$

a. Show that the first four terms of the sequence defined by $a_{n}=n^{2}$ are $$1,4,9,16, \dots$$
b. Show that the first four terms of the sequence defined by $a_{n}=n^{2}+(n-1)(n-2)(n-3)(n-4)$ are also $$1,4,9,16, \dots$$
c. Find a sequence whose first six terms are the same as those of $a_{n}=n^{2}$ but whose succeeding terms differ from this sequence.
d. Find two different sequences that begin $$2,4,8,16, \dots$$

Key Concepts

Geometric Sequences

Geometric sequences are characterized by a constant ratio between successive terms. Recognizing this pattern is essential when encountering sequences with exponential growth, and understanding it enables the exploration of alternative sequences that may share initial terms with a standard geometric progression while diverging later.

Initial Segment Equivalence

This concept highlights that two sequences can coincide for an initial segment of terms even if they are defined by different formulas. By designing functions or adding correction terms that vanish for specific indices, the sequences can be made identical up to a certain point, after which they diverge.

Sequence Modification and Construction

This involves creating new sequences that share certain desired terms with an original sequence while differing in later terms. Techniques include adjusting formulas, adding functions that vanish on a finite set of indices, or using piecewise definitions, which is particularly useful in demonstrating the non-uniqueness of sequences meeting initial conditions.

Sequence Evaluation

This concept involves determining the terms of a sequence by plugging in natural number indices into a given formula. It serves as the foundation for understanding sequences, allowing one to extract information about the behavior of the sequence on a term?by?term basis.

Polynomial Functions

Understanding polynomial functions is crucial, especially when these functions define sequences. Polynomials have properties that allow for manipulating and simplifying expressions, such as recognizing factors that vanish for specific input values. This enables the construction or transformation of sequences while preserving certain terms.

Transcript

00:02 In part a, we are given first four terms of a sequence that is 1 minus 1, minus 1, and so on.

00:10 We need to identify is it a geometric sequence or arithmetic sequence.

00:15 So looking at the terms, i can find the common ratio that is a1 here is 1 and a2 is minus 2.

00:24 So the common ratio must be a2 by a1 which is minus 1 by 1.

00:29 So we get a common ratio of minus 1.

00:31 And checking another term of this sequence that is a3 equals 1 and a4 equals minus 1.

00:38 Again i will try to find our common ratio which is a4 by a3 which is minus 1 by 1 so again i get minus 1.

00:48 Therefore this is a geometric sequence and in order to find next term i will use an ad term formula which is an ad term formula is a n equals a1 r raised to power n minus 1.

01:04 Therefore, next term is 5th.

01:07 So, a5 equals a1 r raised to power 5 minus 1, that is 4.

01:12 So putting the value of a1 and r here, i get 5, a5 equals a1 is 1, r is minus 1 raised to power 4.

01:22 So get 1 into 1, which is 1.

01:25 So my 5 term is a5 which is 1.

01:30 Now in part b, given sequence is root 5 cube root 5 6th root 5 and 1 and so on so firstly first term is root 5 or i can write this as 5 1 1 by 2 and second term is cube root 5 that is 5 raised to power 1 by 3 now looking at the sequence i can recognize this as a geometric sequence so firstly i'll try to find the common ratio so the common ratio is 5 raised to power 1 by 2 1 by 3 by 5 raised to power 1 by 2 as common ratio should be a 2 by a 1 so a 2 is this and a 1 is 5 5 is 2 1 by 2 now using properties of exponents we can write this as 5 5 raise to power 1 by 2 minus 1 by 3 since the base is same in both terms in both of these terms so they can be subtracted so exponential values can be subtracted.

02:46 So we get 1 over 5 raised to power 1 by 6.

02:49 This is our common ratio.

02:51 Now again check for these two terms if our common ratio is same.

02:55 So third term is 5 raise to power 1 by 6, 4 term is 1.

02:59 Therefore our common ratio is a 4 by a 3 which is 1 over 5 to par 1 by 6.

03:06 So yes our common ratio is same.

03:08 So this is a geometric sequence...