HOW TO FIND THE INDICATED TERM FROM GIVEN NTH TERM OF THE SEQUENCE

Example 1 :

Find the indicated terms in each of the sequences whose nth terms are given by

a_n  =  (n + 2)/(2n + 3);     a₇ and a₉

Solution :

In order to find 7^th term and 9^th term, we have to apply 7 and 9 instead of n in the given n^th term of the sequence.

Example 2 :

Find the indicated terms in each of the sequences whose nth terms are given by

a_n  =  (-1)ⁿ 2^{n + 3}(n + 1);     a₅ and a₈

Solution :

In order to find 5^th term and 8^th term, we have to apply 5 and 8 instead of n in the given n^th term of the sequence.

Example 3 :

Find the indicated terms in each of the sequences whose nth terms are given by

a_n  =  2n² - 3n + 1;  a₅ and a₇

Solution :

In order to find 5^th term and 7^th term, we have to apply 5 and 7 instead of n in the given n^th term of the sequence.

Example 4 :

Find the indicated terms in each of the sequences whose nth terms are given by

a_n  =  (-1)ⁿ (1 - n + n²);  a₅ and a₈

Solution :

In order to find 5^th term and 8^th term, we have to apply 5 and 8 instead of n in the given n^th term of the sequence.

Example 5 :

Write an equation for the nth term of the arithmetic sequence. Then find a₂₅.

a)  4, 5, 6, 7, . . .

b) 8, 16, 24, 32, . . .

c) 1, 0, −1, −2, . . .

Solution :

a)  4, 5, 6, 7, . . .

a = 4, d = 5 - 4 ==> 1 and n = 25

a_n = a + (n - 1)d

a₂₅ = 4 + (25 - 1)(1)

= 4 + 24(1)

= 4 + 24

a₂₅ = 28

b) 8, 16, 24, 32, . . .

a = 8, d = 16 - 8 ==> 8 and n = 25

a₂₅ = 8 + (25 - 1)(8)

= 8 + 24(8)

= 8 + 192

a₂₅ = 200

c) 1, 0, −1, −2, . . .

a = 1, d = 0 - 1 ==> -1 and n = 25

a₂₅ = 1 + (25 - 1)(-1)

= 1 + 24(-1)

= 1 - 24

a₂₅ = -23

Example 6 :

Which is the third term of the sequence defined by 

a_n = 4n + 6 ?

a)  6     b)  10     c)  14    d)  18

Solution :

a_n = 4n + 6

To find 3rd term, we have to apply n = 3

a3 = 4(3) + 6

= 12 + 6

= 18

So, option d is correct.

Example 7 :

What is the rule for the n^th term of the arithmetic sequence with a₂₁ = 147 and common difference d = 11 ?

a)  a_n = 11n - 21        b)  a_n = 11n - 42

c)  a_n = 11n + 21       d)  a_n = 11n - 84

Solution :

a_n = a + (n - 1)d

a₂₁ = 147

a = ?, d = 11 and n = 21

147 = a + (21 - 1)11

147 = a + 20(11)

147 = a + 220

a = 147 - 220

a = -73

Applying the value of a and d in the formula of nth term, we get

a_n = -73 + (n - 1)(11)

= -73 + 11n - 11

= -84 + 11n

a_n = 11n - 84

So, option d is correct.

Example 8 :

Find the indicated term for each of the following sequences.

T₂₁ = 195, T₁₀ = 85 and T₁

Solution :

T₂₁ = 195

a + 20d = 195 -------(1)

T₁₀ = 85

a + 9d = 85 --------(2)

(1) - (2)

a + 20d - (a + 9d) = 195 - 85

20d - 9d = 110

11d = 110

d = 110/11

d = 10

Applying the value of d, we get

a + 9(10) = 85

a + 90 = 85

a = 85 - 90

a = -5

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