Question 1: If are in A.P., prove that:
i) are in A.P.
ii) are in A.P.
Answer:
i) Given
To prove:
Hence proved.
ii) Given
To prove:
Question 2: If are in A.P., prove that
are in A.P.
Answer:
Given are in A.P.
are in A.P.
are in A.P.
are in A.P.
are in A.P.
Question 3: If are in A.P., then show that:
i) are also in A.P.
ii) are in A.P.
iii) are in A.P.
Answer:
i) Given are in A.P. Therefore
To prove:
Hence proved.
ii) Given are in A.P. Therefore
To prove:
Hence proved.
iii) Given are in A.P. Therefore
To prove:
RHS
LHS. Hence proved.
Question 4: If are in A.P., prove that
i) are in A.P.
ii) are in A.P.
Answer:
i) Given are in A.P.
are in A.P.
ii) From i) since are in A.P.
. Hence proved.
Question 5: If are in A.P., then prove that:
i)
ii)
iii)
Answer:
i) Given are in A.P.
To prove:
RHS
LHS Hence proved.
ii) Given are in A.P.
To prove:
RHS
LHS. Hence proved.
iii) Given are in A.P.
To prove:
RHS
LHS. Hence proved.
Question 6: If are in A.P., prove that
are in A.P.
Answer:
Given are in A.P.
are in A.P.
are in A.P.
Dividing by we get
are in A.P.. Hence proved.
Question 7: Show that and
are consecutive terms of an A.P., if
are in A.P.
Answer:
Given are in A.P..
Therefore
To prove: are in A.P.
LHS
Therefore and
are consecutive terms of an A.P.