Write the first five terms in each of the following:
Question:

Let < an > be a sequence. Write the first five terms in each of the following:

(i) a1 = 1, an = an − 1 + 2, n ≥ 2

(ii) a1 = 1 = a2an = an − 1 + an − 2n > 2

(iii) a1 = a2 = 2, an = a− 1 − 1, n > 2

Solution:

(i) $a_{1}=1, a_{n}=a_{n-1}+2, n \geq 2$

$a_{2}=a_{1}+2=1+2=3$

$a_{3}=a_{2}+2=5$

$a_{4}=a_{3}+2=7$

$a_{5}=a_{4}+2=9$

Hence, the five terms are 1, 3, 5, 7 and 9.

(ii) a1 = 1 = a2an = an − 1 + an − 2n > 2

$a_{3}=a_{2}+a_{1}=1+1=2$

$a_{4}=a_{3}+a_{2}=2+1=3$

$a_{5}=a_{4}+a_{3}=3+2=5$

Hence, the five terms are 1, 1, 2, 3 and 5.

(iii) a1 = a2 = 2, an = a− 1 − 1, n > 2

$a_{3}=a_{2}-1=2-1=1$

$a_{4}=a_{3}-1=1-1=0$

$a_{5}=a_{4}-1=0-1=-1$

Hence, the five terms are $2,2,1,0$ and $-1$.

 

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