Need help with this equation n2.(a) Find the first four terms of the sequence defined by an%Da1%Da2%Da3a4n2 + (n – 1)(n – 2)(n – 3)(n – 4) are the same as%3D(b) Show that only the first four terms of the sequence defined by dnan= n2 by finding the first five terms of d, and a5.did2d3d4d5a5n2 but whose succeeding terms differ from this(c) Find a sequence whose first six terms are the same as those of ansequence.an%3D(d) The first four terms of the sequence an = 3" are3, 9, 27, 81, ..Find another sequence that has the same first four terms.dn :III| ||II

Answered: (a) Find the first four terms of the…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.3: Geometric Sequences

Problem 82E

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Need help with this equation

n2.
(a) Find the first four terms of the sequence defined by an
%D
a1
%D
a2
%D
a3
a4
n2 + (n – 1)(n – 2)(n – 3)(n – 4) are the same as
%3D
(b) Show that only the first four terms of the sequence defined by dn
an
= n2 by finding the first five terms of d, and a5.
di
d2
d3
d4
d5
a5
n2 but whose succeeding terms differ from this
(c) Find a sequence whose first six terms are the same as those of an
sequence.
an
%3D
(d) The first four terms of the sequence an = 3" are
3, 9, 27, 81, ..
Find another sequence that has the same first four terms.
dn :
II
I| ||
II

Expert Solution

Step 1

(a) A sequence is given by $a_{n} = n^{2}$

We need to find first four terms of the sequence.

$a_{1} = 1^{2} = 1 a_{2} = 2^{2} = 4 a_{3} = 3^{2} = 9 a_{4} = 4^{2} = 16$

Hence, first four terms are:

$a_{1} = 1 a_{2} = 4 a_{3} = 9 a_{4} = 16$

Step 2

(b) Now, given sequence is $d_{n} = n^{2} + (n - 1) (n - 2) (n - 3) (n - 4)$

Finding first five terms of this sequence,

$\begin{array}{rcl} d_{1} & = & 1^{2} + (1 - 1) (1 - 2) (1 - 3) (1 - 4) \\ = & 1 + 0 \\ = & 1 \\ d_{2} & = & 2^{2} + (2 - 1) (2 - 2) (2 - 3) (2 - 4) \\ = & 4 + 0 \\ = & 4 \\ d_{3} & = & 3^{2} + (3 - 1) (3 - 2) (3 - 3) (3 - 4) \\ = & 9 + 0 \\ = & 9 \\ d_{4} & = & 4^{2} + (4 - 1) (4 - 2) (4 - 3) (4 - 4) \\ = & 16 + 0 \\ = & 16 \\ d_{5} & = & 5^{2} + (5 - 1) (5 - 2) (5 - 3) (5 - 4) \\ = & 25 + 4 \cdot 3 \cdot 2 \cdot 1 \\ = & 25 + 24 \\ = & 49 \end{array}$

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