2. Use the Geometric Sum Formula to do the following. (a) Evaluate 1 +3+3² +3³ + · ·.+310. (b) (Part a, P20, Page 21) Find a formula for 1 1 1 + 1+x² (1+x²)² (1+x²)" *

Q: Use iteration to guess an explicit formula for the sequence: Tk = rk-1+3· 5* for all integers k > 1;…

A: Explained below

Q: The simple formula for a, if the first 5 terms of the sequence {an} are 1, 21, 121, 621, 3121 is:…

A: We are asked to find which is correct for the sequence.

Q: 8. Give the first five terms of the following recursively defined sequence: a1 = 2. and ar = ax-1 +…

A: Recursive definition of a sequence is one in which one term can be determined using previous terms…

Q: Use the approach in Gauss’s problem to find the following sums of arithmetic sequences (do not use…

A: To find the sum of the given arithmetic sequence using Gauss's approach.The given sequence is…

Q: i. Write out the 1st four terms of the sequence {1+(-1)"}o ii. Use the result from part (i) to…

A:

Q: Write out the 1st four terms of the sequence {1+(-1)"}"o In=D0 Use the result from part (i) to…

A:

Q: Propose a formula for the sum of the first n terms of the sequence. Then use mathematical induction…

A:

Q: 1 The sequence an %3D 2n +3 (a)decresing (b) in

A: See the details solution in below

Q: 4n2 + 2n + 6 6, 15, 24, ...,

A:

Q: Use sigma notation to represent the sum of the first seven terms of the following sequence 4,10,16

A: Given sequence, 4,10,16,... To find the sigma notation for the first seven terms of the sequence.

Q: Use summation notation to Write the senes 3(1) 3(2) 3(3) 3(0) The summation notation for the series…

A: Answer is given below.

Q: What is the summation from n = 1 to infinity, of  (3n+1) / (5n + 4) ?

A:

Q: 5 Compute the sum: 3n

A: Take 3 out of the summation and the use the formula for sum of n natural numbers.

Q: Simplify by comnbining the two sums and find an explicit formula. That is, your answer should not…

A:

Q: 11. Consider the sequence of sums 1, 1 + 2, 1 + 2 + 3, 1+2+3+4,1+2+3+4+5, .. (a) What happens to the…

A: Consider the sequence of sums 1, 1+2, 1+2+3, 1+2+3+4,.....  a) What happens to the terms of this…

Q: 4. Define the sequence an by a, = 1, a, = 33, and an = 3an-1+ 10an-2 for n 2 3. Prove that an = 5" –…

A: Concept used :- we use mathematical induction to solve the given question.

Q: 25 Evaluate the sum: 4k²) - k=1

A: Explanation of the solution is given below..

Q: Verify that the geometric sum formula gives the correct result for the sums 1 + (1)/(2) and (1)/(2)+…

A: Formula for the sum of a geometric  series:Sn=a11-rn1-r    r≠1a1=the first termn=number of…

Q: 1) Using the appropriate formula, find the 17th term of the sequence 7, 4, 1,.... 2) If h is defined…

A:

Q: (nt) n=1

A: we have to find the sum

Q: Express the following sum in closed form. What is n (ending term)? ∑nk=1(5k−3)=

A: Consider the given series.Then apply the sum rule on the expression.

Q: Find an example of a sequence with the given property. a) xn - 3 and x, # 3 for all n. - 3 and the…

A:

Q: A recursive formula for the sequence 18, 9, 4.5, ... is 1 91 = 18; gn = = In-1 1. 9n = 18( -1 2. 91…

A: Given- g1=18,gn=12gn-1 Putting n=2, g2=12g1=12×18=9 Putting n=3, g3=12g1=4.5 According to this…

Q: The Fibonacci sequence is defined as an=1,1,2,3,5,8,13,....  Where a1=a2=1 & an=an-1+an-2 Prove by…

A: Given: The Fibonacci sequence is defined as an=1,1,2,3,5,8,13,.....where, a1=a2=1 & an=an-1+an-2…

Q: What is the formula for the nth term tn of the folowing sequence tn = -4 + tn-1 when t3 = pi/6,…

A: Given tn= -4+tn-1  ---(1)and    t3 = π6

Q: We define a sequence by b = 1, b2 = 1, and b, = 2bn-1 + bn-2 for n2 3. Use strong mathematical…

A:

Q: 08. The sequence { GT) }n-1 3n+2 37+1

A:

Q: Use the rules for sums of powers of integers to compute the sum. 21 Σ k k=3

A:

Q: Prove { (8n-7)/n} n=1 to infinity is an increasing sequence.

A: If we take the sequence as {Un}, and if we can able to show that for all n=1,2,3, ...  Un+1>Un…

Q: 6. Find the first five terms, ao,..., a4, of the following recursively defined sequence: ak = k?…

A:

Q: Use the rules for sums of powers of integers to compute the sum. 24 4.² k=8

A: Recall: sum of squares of n integer                    ∑i=1ni2=n(n+1)(2n+1)6

Q: Use sigma notation to write the sum. [금)-(금)·. ()-끝() 9 (4) 3 3 3n 3n 3 + n Σ i = 1

A: Look at the question there is a variable multiplied with 3 inside the square bracket which is…

Q: Express the following sum in closed form. Σ 2 k > (4 +3 in k=1 -(+-2 ). k Hint: Start by multiplying…

A:

Q: 3. Consider a sequence co, C1, C2, .. . defined recursively Ck = 3 Ck-1 +1 for all k >1 and co = 2.…

A: Given recurrence relation is ck=3ck-1+1 , k≥1             (i) Given initial condition is c0=2.…

Q: Use Gauss' trick to determine the sum 1+3 + 5 +... + (2n – 3) + (2n – 1) + (2n + 1

A: Gauss's trick is very significant in determining the sum of a series. This method includes the…

Q: 2. Find the first five terms of these recursively defined sequence. a1 = 2; an+1=3an-5

A:

Q: i. Write out the 1ª four terms of the sequence {1 +(-1)"}n=o ii. Use the result from part (i) to…

A: For (i), We know that (-1)n =1 for even values of n and (-1)n=-1 for odd values of n. Thus, the…

Q: Q5. i. Write out the 1ª four terms of the sequence {1+(-1)"}"=o In=0 ii. Use the result from part…

A: i) given sequence {1+(-1)n}n=0+∞ for n=0  1+(-1)0  = 1 +1 =2  for n=1 1+(-1)1  = 1 -1 =0 for n=2…

Q: List the first five terms of the sequence. (-1)^-1 3n 21 co 32 а3 = а4 - 25 = - an - XXXXX Х Х Х

A:

Q: 1. Give a recursive definition of a sequence an, n > 1, where an 5n – 7

A: Please see the next step for solution.

Q: 3) Find three different sequences beginning with the terms 3, 5, 7 whose terms are generated by a…

A:

Q: Find the sum of the infinite series. 1 1 1 6-18 18- 30 30-42 +... 42-54 (Use symbolic notation and…

A:

Q: an = {-5,–1,3,7,11, ... }

A:

Q: 3. Consider a sequence co, C1, C2, . . . defined recursively Ck = 3 Ck-1 + 1 for all k >1 and co =…

A: Here, the given recursive formula is ck=3ck-1+1 such that c0=2

Q: A test question lists the first four terms of a sequence as 2, 4, 6, and 8 and asks for the fifth…

A: Given sequence is: 2,4,6,8, Here we observe that: 4-2=26-4=28-6=2 Thus, we observe that each…

Q: The generating function of the sequence 3,3,3,3,3 is 1-x5 (1-x) 1-3x° O 11-x) 1-27x (1-x) 1-243x…

A:

Q: Use the approach in Gauss's Problem to find the following sums of arithmetic sequences.  A.…

A: Consider the series 1+2+3+4+…+1001. For the series 1+2+3+4+…+1001, it is clear that there are 1001…

Q: 4 24. (_^ -ン Sn %3D

A:

Q: Determine the boundlessness and monotonicity of the sequence An 2n+1 %3D 3n-2

A:

Q: Use sigma notation to write the sum. [6 - ()°] + [6 - (4)°] +-- • [s - (2)'] k = 1

A: According to the problem, we have