Question 4, Let {an} be a sequence of real number. n starts from 1. Let a1 =1 and for any n 2 2, we have a, = 2a,-1+1. (a) Find an explict formula for an, (Hint consider b, = an +1, b, should be a sequence with nice property) (b) Find the partial sum, Sn, of an, ie Sn =E=1ª;. %3D %3D

Q: an 2. (14 points) Use mathematical induction to prove that a,, = -5+n(n+1), -8 for all integers n >…

A: To prove that 𝑎n = −5+𝑛(𝑛+1) for all integers 𝑛>1 is an explicit formula for the sequence…

Q: Let (an) 1 and (bn)-1 be two sequences of real numbers. Prove or disprove each of the following…

A: “Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: 10. (6 points) A sequence is given by the recursive definition an = an-1 + 2, ao = 3. a) Find a₁,…

A: The objective of the question is to find the first three terms of the sequence defined by the…

Q: 11. (6 points) A sequence is given by the recursive definition bn = 4bn-1, bo = 3. a) Find b₁,b₂,…

A: The objective of the question is to find the values of b_1, b_2, b_3 and a general formula for b_n,…

Q: Enter the number to complete the linear combination. gcd(57, 39) yields sequence: 57 39 18 30 18 57…

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Q: 5. Consider the sequence defined by the recursive formula AN = 2 AN-1-3 AN-2 and starting with A₁ =…

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Q: Write the first five terms of the recursively defined sequence. 12 ak 2² a₁ = 4₁ ak + 1 Step 1 Note…

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Q: A sequence {a,} is defined recursively by aj = 1, az = 4, a3 = 9 and an = an–1 – an–2+ an-3 + 2(2n –…

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Q: 4. Consider the sequence 8 1 {2(k+3 (2(k+3)) k=1 rewrite the sequence from j-5

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Q: 4 8 Suppose the sequence {b1, b2, b3, ...} is given by {-1, , 9' 27 Which, if any, of the following…

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Q: The number sequence a1, a2, a3, .... is defined by a1 = 8 a2 = 24 and recursion formula 4an + 1…

A: We need to show that an=5n-12n-2 for a1=8 and a2=24  The given recursion formula is…

Q: 200 ra, is 25a 200 200 Let (1+ x+ x*) = Ïq,x and a = Ea, , then value of

A: We will use basic trics to arrive at the solution

Q: For the sequence an=-2an-1 and ao = -1, the values of the first six terms are ao = and a5 = a4= , a₁…

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Q: Prove by induction Bernoulli’s Identity: For every real number x > −1 and every positive integer n,…

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Q: tion 1 1) Write the next three terms of the following sequences. 3 4 5 Ка) 2, 4'9'16 ' ... a1 =2 (b)…

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Q: 36 Find the functian that represents the sequence shown below. V= {-2, 6, - 18, 54, - 162, .} O pla)…

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Q: Find the explicit formula for the geometric sequence b, with the following information given. bị = 7…

A: Given : b1=7b4=-5627

Q: O List the first five terms of the sequence 2n +(-3)" }

A: The given sequence is: -1n+12n+-3n To find the first five-term of the sequence.

Q: Which of the following is the 5th term of the sequence an = (-1)^n /(2n-1)   1. 1/9   2.  -1/10…

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Q: Show that if p is a prime number, a and b are positive integers and p | ab, then p | a or p | b.

A: The objective is to prove that if p is a prime number, a and b are positive integers and p | ab,…

Q: and (bn) be two sequences of real number disprove each of the following statements: n=1 n=1 ¹) If…

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Q: Example 4. How many terms of the series 1, 6, 11, sum is 148, be taken so that their

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Q: ! 3) Determine eat for anthmeric sequence a = -15, 9, 82-85 SI-A PISK.C@ $10.011X

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Q: 12. Find the first 5 terms of the sequence: 2n a, = (-1)" а) 2 n² +1 b) a, = 4a - 5 n-1 and ª1 =1

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Q: Write an expression for the apparent nth term of the sequence , -, - 16 an: (-1)"+1 an = 2h -2n an =…

A: Observe the pattern in the given sequence to find the nth term.

Q: + 2". Find The sequence {a, } is given by the formula a, = [2n] + a) ao = b) a1 = c) az = d) az =

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Q: vīz, vi2+ V12 , V12+ V12+ V12 ,. .... Consider this sequence: i.e., a, = V12, a,1 = V12+a, ,n>1 Use…

A: Given a1=12 and an+1=12+an, n>1 First of all, we will write the first 10 terms in the given…

Q: Write an explicit formula for am, the nth term of the sequence 35, 38, 41, ...

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Q: २ Let (an) a sequence that satisfies a³ + 6 for n = 1, 2, ... Manti If a₁ = 2/1/2, show that and…

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Q: A sequence is said to be decreasing sequence if

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Q: 6. Let {an}1 be a sequence. If a1 = 1 and An+1 :2an + 1, use induction to prove an = 2" – 1. %3|

A: given that:-              an =1 and an+1 =2an+1  now, to prove  an=2n-1 we use induction.

Q: The sum of the first n terms of a sequence is S₁₁ = 2 - n a) Determine the 1st and 2nd term of this…

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