Given several terms of a sequence {a_n}^infinity n=1 complete parts a. through c

Question

Given several terms of a sequence {a_n}^infinity n=1 complete parts a. through c. below {9, -9.9, -9,9, ..} Find the next two terms of the sequence The next two terms are a_6 = -9 and a_7 = 9. Find a recurrence relation that generates the sequence (supply the initial value of the in a_n+1 =1 -a_n. a_1 =9for ngreaterthanorequalto 1 a_n+1 =1 -a_n. a_1 =9for ngreaterthanorequalto 1 Find an explicit formula for the general or nth term of the sequence | The nth term of the sequence is equal to na_n+1 =1 -a_n. a_1 =9for ngreaterthanorequalto 1

Explanation / Answer

Given that

The sequance is , { 9 , -9 , 9 , -9 , 9 , ......... }

b ) The recurrence relation that generates the sequaence is ,

an+1 = -an , a1 = 9 for n 1

   n = 1

a1+1 = -a1

   a2 = -9

   n = 2

a2+1 = -a2

a3  = -(-9)

a3   = 9

      n = 3

a3+1 = -a3

   a4 = -9

n = 4

a4+1 = -a4

   a5 = -(-9)

a5 = 9

c ) We know that

A geometric sequence has a constant ratio r and is defined by an = a0.rn-1

Compute constant ratio r = an/an-1

   n=1

r = a1/a1-1 = a1/a0 = 9/-9 = -1

r = -9/9 = -1

r = 9/-9 = -1

r = -9/9 = -1

The ratio of all adjacent terms is same and equal to r = -1

The first term of sequence is a0 = 9

The nth term of the sequance is , an = a0.rn-1

an = 9.(-1)n-1, n 1

Therefore,

  The nth term of the sequance is , an = 9.(-1)n-1 ,n 1

a )   an = 9.(-1)n-1 ,n 1

a6 = 9.(-1)6-1

   a6 = 9.(-1)

   a6 = -9

  an = 9.(-1)n-1 ,n 1

a7 = 9.(-1)7-1

   a7 = 9.(1)

   a7 = 9

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