Sequence and series Example

Question 1: If 4,7,10,13,16,19,22……is a

         sequence, Find:

 Ⓐ Common difference

 Ⓑnth term

 Ⓒ21st term

Solution: Given sequence is, 4,7,10,13,16,19,22……

Ⓐ) The common difference 

d=A2-A1

 = 7 – 4

= 3

Ⓑ) The nth term of the arithmetic sequence is denoted by the term An and is given by An = A₁ + (n-1)d, where “A” is the first term and d is the common difference.

An = 4 + (n – 1)3 

    = 4 + 3n – 3 

    = 3n + 1

c) 21st term as:  

An = A1 + (n-1)d

A21 = 4 + (21-1)3 

     = 4+60 

     = 64

Question 2: Consider the sequence 1, 4, 16, 64, 256, 1024….. Find the common ratio and 9th term.

Solution: 

The common ratio 

    r=G2/G1

    r= 4/1 

     = 4

The preceding term is multiplied by 4 to obtain the next term.

The nth term of the geometric sequence is denoted by the term Gn and is given by Gn = r^-¹G1

where a is the first term and r is the common ratio.

Here G1= 1, r = 4 and n = 9

So, 9th term is can be calculated as

     Gn = r^-¹G1

     G9 = (4)⁹-¹(1)

        = 4⁸ 

        = 65536

Learning about mathematical concepts can be so much fun😁😁.

3. The sum of three numbers which

   are consecutive terms of an A.P. is

   21. If the second number is reduced

   by 1 while the third is increased by 1,

   three consecutive terms of a G.P.

   result. The three numbers is

Ⓐ) 3,7, 11     Ⓒ)1,7,11

Ⓑ) 12, 7, 2    Ⓓ) a and b

solution:

Let the three numbers in A.P. is

A-B, A, A+B

B=reduce 1 or increase 1and A is require number.

   A-B, A, A+B

Given፦ (A-B)+A+(A+B) = 21

                  3a =21

                   a= 21÷3 

                    =7

The numbers are፦7-B, 7,7+B

If the second number is reduced by 1 while the third number is increased by 1, the resulting numbers are 7- B, 6, 8+B

which are given to be in G.P

    6/7-B=8+B/6

36= (7-B) (8 + B)

36= 56-B+B²

B²-B+20

(B-5) (B+4)=0

B=-5, B=4

 When B=-5;

           7-B, 7,7+B

           7-(-5),7,7+(-5)

           7+5,7,7-5

            12,7,2

The numbers are 12, 7, 2 and

  When B= 4,

          7-B, 7,7+B

          7-4,7,7+4

           3,7,4

When d= 4, the numbers are 3, 7, 11.

Answer፦d

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