If the sum of first ‘p’ terms of an A.P. is ‘q’

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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.  = d

S_p = \dfrac{p}{2}[2a + (p-1)d]

\Rightarrow q = \dfrac{p}{2}[2a + (p-1)d]

\Rightarrow 2q = 2ap + p(p-1)d  .  .  . (i)

Similarly,

S_q = \dfrac{q}{2}[2a + (q-1)d]

\Rightarrow p = \dfrac{q}{2}[2a + (q-1)d]$ [latex]\Rightarrow 2p = 2aq + q(q-1)d  .  .  . (ii)

Substracting eq (ii) to (i)

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

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

⇒   2(p – q) = 2a(q – p) + (q² – q – p² + p)d

⇒  2(p – q) = 2a(q – p) + {(q² – p²) -(q – p)}d

⇒ 2(p – q) =  2a(q – p) + {(q – p)(q + p) -(q – p)}d

⇒ 2(p – q) = (q – p){2a + (q  + p -1)d}

⇒  2(p-q) = -(p – q){2a + (p + q – 1)d}

⇒  2 = – {2a + (p + q – 1)d} .  .  . (iii)

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

S_{(p + q)} = \dfrac{(p + q)}{2}[2a + (p + q - 1)d]

= \dfrac{(p + q)}{2}[2a + (p + q - 1)d] [From eq (iii)]

= \dfrac{(p + q)}{2}[-2]

= -(p+q)

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

Solution: For solution click here

Question 2:

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

Solution: For solution Click Here

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]

Solution: For solution click here

Question 4:

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

Solution : For solution click here

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.                          [CBSE  2019]

Solution: for solution click here

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 \dfrac{(a + c)(b + c -2a)}{2(b-a)}.                                                    [CBSE      2020]

Solution: For solution click here

Question 8:

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

Answer : For solution click here

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.                           [CBSE   2017]

Solution: For solution click here

Question 10:

A sum of ₹ 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 ₹ 50 less than its preceding prize, find the value of each of the prizes.

Solution: For solution click here

 

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