Advertisements
Advertisements
प्रश्न
The sum of the first four terms of an A.P. is 56. The sum of the last four terms is 112. If its first term is 11, then find the number of terms.
Advertisements
उत्तर
Let the A.P. be a, a + d, a + 2d, a + 3d, ... a + (n – 2) d, a + (n – 1)d.
Sum of first four terms = a + (a + d) + (a + 2d) + (a + 3d) = 4a + 6d
Sum of last four terms = [a + (n – 4) d] + [a + (n – 3) d] + [a + (n – 2) d]
+ [a + n – 1) d]
= 4a + (4n – 10) d
According to the given condition,
4a + 6d = 56
⇒ 4(11) + 6d = 56 [Since a = 11 (given)]
⇒ 6d = 12
⇒ d = 2
∴ 4a + (4n –10) d = 112
⇒ 4(11) + (4n – 10)2 = 112
⇒ (4n – 10)2 = 68
⇒ 4n – 10 = 34
⇒ 4n = 44
⇒ n = 11
Thus, the number of terms of the A.P. is 11.
संबंधित प्रश्न
If the sum of first p terms of an A.P. is equal to the sum of the first q terms, then find the sum of the first (p + q) terms.
If the sum of n terms of an A.P. is 3n2 + 5n and its mth term is 164, find the value of m.
The pth, qth and rth terms of an A.P. are a, b, c respectively. Show that (q – r )a + (r – p )b + (p – q )c = 0
Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in annual installment of Rs 1000 plus 10% interest on the unpaid amount. How much will the scooter cost him?
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely real ?
Find the second term and nth term of an A.P. whose 6th term is 12 and the 8th term is 22.
The sum of three terms of an A.P. is 21 and the product of the first and the third terms exceeds the second term by 6, find three terms.
Find the sum of the following arithmetic progression :
1, 3, 5, 7, ... to 12 terms
Find the sum of the following arithmetic progression :
41, 36, 31, ... to 12 terms
Find the sum of the following serie:
101 + 99 + 97 + ... + 47
Find the sum of all integers between 84 and 719, which are multiples of 5.
Find the sum of all even integers between 101 and 999.
The number of terms of an A.P. is even; the sum of odd terms is 24, of the even terms is 30, and the last term exceeds the first by \[10 \frac{1}{2}\] , find the number of terms and the series.
If Sn = n2 p and Sm = m2 p, m ≠ n, in an A.P., prove that Sp = p3.
If a, b, c is in A.P., then show that:
b + c − a, c + a − b, a + b − c are in A.P.
If a, b, c is in A.P., prove that:
a3 + c3 + 6abc = 8b3.
If x, y, z are in A.P. and A1 is the A.M. of x and y and A2 is the A.M. of y and z, then prove that the A.M. of A1 and A2 is y.
A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual instalments of Rs 500 plus 12% interest on the unpaid amount. How much the tractor cost him?
A man is employed to count Rs 10710. He counts at the rate of Rs 180 per minute for half an hour. After this he counts at the rate of Rs 3 less every minute than the preceding minute. Find the time taken by him to count the entire amount.
We know that the sum of the interior angles of a triangle is 180°. Show that the sums of the interior angles of polygons with 3, 4, 5, 6, ... sides form an arithmetic progression. Find the sum of the interior angles for a 21 sided polygon.
In a cricket team tournament 16 teams participated. A sum of ₹8000 is to be awarded among themselves as prize money. If the last place team is awarded ₹275 in prize money and the award increases by the same amount for successive finishing places, then how much amount will the first place team receive?
Write the common difference of an A.P. the sum of whose first n terms is
If log 2, log (2x − 1) and log (2x + 3) are in A.P., write the value of x.
Write the sum of first n odd natural numbers.
If the sum of n terms of an A.P. is 2 n2 + 5 n, then its nth term is
If a1, a2, a3, .... an are in A.P. with common difference d, then the sum of the series sin d [cosec a1cosec a2 + cosec a1 cosec a3 + .... + cosec an − 1 cosec an] is
If a1, a2, a3, .... an are in A.P. with common difference d, then the sum of the series sin d [sec a1 sec a2 + sec a2 sec a3 + .... + sec an − 1 sec an], is
Mark the correct alternative in the following question:
\[\text { If in an A . P } . S_n = n^2 q \text { and } S_m = m^2 q, \text { where } S_r \text{ denotes the sum of r terms of the A . P . , then }S_q \text { equals }\]
The first term of an A.P. is a, the second term is b and the last term is c. Show that the sum of the A.P. is `((b + c - 2a)(c + a))/(2(b - a))`.
The pth term of an A.P. is a and qth term is b. Prove that the sum of its (p + q) terms is `(p + q)/2[a + b + (a - b)/(p - q)]`.
The first term of an A.P.is a, and the sum of the first p terms is zero, show that the sum of its next q terms is `(-a(p + q)q)/(p - 1)`
A man accepts a position with an initial salary of Rs 5200 per month. It is understood that he will receive an automatic increase of Rs 320 in the very next month and each month thereafter. Find his salary for the tenth month
If the sum of n terms of an A.P. is given by Sn = 3n + 2n2, then the common difference of the A.P. is ______.
If 9 times the 9th term of an A.P. is equal to 13 times the 13th term, then the 22nd term of the A.P. is ______.
If in an A.P., Sn = qn2 and Sm = qm2, where Sr denotes the sum of r terms of the A.P., then Sq equals ______.
Let Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn then S3n: Sn is equal to ______.
Let 3, 6, 9, 12 ....... upto 78 terms and 5, 9, 13, 17 ...... upto 59 be two series. Then, the sum of the terms common to both the series is equal to ______.
