Question

1.)Check weather 0 is a term of the AP 31,28,25,..................? 2.)justify your answer if the numbers n-2,4n-1 ,5n+2are in AP find the value of n?

Answer 1

Answer 1):

Necessary condition for any sequence to be in arithmetic progression (AP) is that difference of consecutive terms must be equal.

So let's check if we are getting same difference or not

28-31=-3

25-28=-3

Each time we are getting same common difference -3.

Hence given sequence is in AP and common difference d=-3

first term a= 31

Now we have to check if 0 is any term of that sequence or not so let's plug these values into nth term formula

an=a+(n-1)d

0=31+(n-1)(-3)

0=31-3n+3

0=34-3n

3n=34

n=34/3

We know that term can't be fraction.

Hence 0 is NOT a term of given sequence.

Answer2):

If given sequence say a,b,c is in AP then we can use formula 2b=a+c

For given sequence a=n-2, b=4n-1, c=5n+2

plug these values into above formula

2(4n-1)=(n-2)+(5n+2)

8n-2=n-2+5n+2

8n-2=6n

2n-2=0

2n=2

n=1

now plug value of n=1 into given terms to find their values

n-2=1-2=-1

4n-1=4*1-1=4-1=3

5n+2=5*1+2=5+2=7

-1,3,7 differ by 4 so yes they are in AP.

Hence final answer is n=1