# Question 6:

If the sum of first ‘p’ terms of an A.P. is ‘q’ and the sum of first ‘q’ terms is ‘p’ ; then show that the sum of the first (p + q) terms is {-(p + q)}.

## Solution:

Let the first term of an A.P. = a

Common difference of A.P.Â  = d

Â  .Â  .Â  . (i)

Similarly,

Â  .Â  .Â  . (ii)

Substracting eq (ii) to (i)

2p – 2q = [2aq + q(q – 1)d] – [2ap + p(p – 1)]

â‡’ 2(p – q) = 2aq + q(q – 1)d – 2ap – p(p – 1)

â‡’Â  Â 2(p – q) = 2a(q – p) + (qÂ² – q – pÂ² + p)d

â‡’Â  2(p – q) = 2a(q – p) + {(qÂ² – pÂ²) -(q – p)}d

â‡’ 2(p – q) =Â  2a(q – p) + {(q – p)(q + p) -(q – p)}d

â‡’ 2(p – q) = (q – p){2a + (qÂ  + p -1)d}

â‡’Â  2(p-q) = -(p – q){2a + (p + q – 1)d}

â‡’Â  2 = – {2a + (p + q – 1)d} .Â  .Â  . (iii)

Now, sum of (p + q) terms of an A.P.

[From eq (iii)]

Hence, of (p + q) terms of A.P.Â  = – (p + q)

# Question 1:

Ramkali would require Rs 5000 for getting her daughter admitted in a school after a year. She saved Rs 150 in the first month and increased her monthly by Rs 50 every month. Find, if she will be able to arrange the required money after 12 months. Which value is reflected in her efforts ?

# Question 2:

The first term of an A.P. is -5 and the last term is 45. If the sum ofÂ  the terms of the A.P. is 120, then find the number of terms and the common difference.

# Question 3:

If the sum of the first 6 terms of an A.P. is 36 and that of the first 16 terms is 256, find the sum of the first 10 terms.

[CBSE 2016, 13, 12]

# Question 4:

The digits of a positive number of three digits are in A.P.Â  andÂ  their sum is 15. The number obtained by reversing the digitsÂ  is 594 less than the original number. Find the number.

# Question 5:

If the sum of first m terms of an A.P. is some as the sum of its first n terms, show that the sum of its first (m + n) terms is zero.Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  [CBSEÂ  2019]

# Question 7:

Show that the sum of all terms of an A.P. whose first term is a, the second term is b and the last term is c is equal to .Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  [CBSEÂ  Â  Â  2020]

# Question 8:

Yasmeen saves â‚¹ 32 during the first month â‚¹ 36 in the second month and â‚¹ 40 in the third month. If she continues to save in this manner, in how many months will she save â‚¹ 2000?

# Question 9:

A child puts one five-rupees coin of her savings in the piggy bank on the first day. She increases her saving by one five-rupee coin daily. If the piggy bank can hold 190 coins of five rupees in all, find the number of days she can continue to put the five -rupee coins into it and find the total money she saved.Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â [CBSEÂ  Â 2017]

# Question 10:

A sum of â‚¹ 4,250 is to be used to give 10 cash prizes to students of a school for their overall academic performance. If each prize is â‚¹ 50 less than its preceding prize, find the value of each of the prizes.