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.