Answer:
(a)= option 3
(b)= option 3
Step-by-step explanation:
(A) It is given that a sequence is defined by the recursive function [tex]a_{1}=1[/tex] and [tex]a_{n}=a_{n-1}+n[/tex]
Now, substituting the value of n=2 in above equation,
[tex]a_{2}=a_{1}+2[/tex]
[tex]a_{2}=1+2[/tex]
[tex]a_{2}=3[/tex]
Again putting n=3,
[tex]a_{3}=a_{2}+3[/tex]
[tex]a_{3}=3+3[/tex]
[tex]a_{2}=6[/tex]
Putting n=4,
[tex]a_{4}=a_{3}+4[/tex]
[tex]a_{4}=6+4[/tex]
[tex]a_{4}=10[/tex]
Putting n=5,
[tex]a_{5}=a_{4}+5[/tex]
[tex]a_{5}=10+5[/tex]
[tex]a_{5}=15[/tex]
Putting n=6,
[tex]a_{6}=a_{5}+6[/tex]
[tex]a_{6}=15+6[/tex]
[tex]a_{6}=21[/tex]
Putting n=7,
[tex]a_{7}=a_{6}+7[/tex]
[tex]a_{7}=21+7[/tex]
[tex]a_{7}=28[/tex]
Hence, option 3 is correct.
(B) The given sequence is :
5, -10, 20, -40, 80.....
Since, the given sequence is GP, therefore
[tex]a_{1}=5[/tex] and r=-2
The nth term is given by:
[tex]a_{n}=a_{1}r^{n-1}[/tex]
For fifteenth term, put n=15 in above equation, we get
[tex]a_{15}=5(-2)^{14}[/tex]
=[tex]5{\times}16384[/tex]
=[tex]81920[/tex]
Hence, the fifteenth term is 81920.
Option 3 is correct.
