Advertisements
Advertisements
प्रश्न
How many terms are there in the A.P. whose first and fifth terms are −14 and 2 respectively and the sum of the terms is 40?
Advertisements
उत्तर
\[\text { We have: } \]
\[ a = - 14 \text { and } S_n = 40 . . . (i)\]
\[ a_5 = 2\]
\[ \Rightarrow a + \left( 5 - 1 \right)d = 2\]
\[ \Rightarrow - 14 + 4d = 2\]
\[ \Rightarrow 4d = 16\]
\[ \Rightarrow d = 4 . . . (ii)\]
\[\text { Also }, S_n = \frac{n}{2}\left[ 2a + (n - 1)d \right]\]
\[ \Rightarrow 40 = \frac{n}{2}\left[ 2\left( - 14 \right) + (n - 1) \times 4 \right] (\text { From }(i) \text { and } (ii))\]
\[ \Rightarrow 80 = n\left[ - 28 + 4n - 4 \right]\]
\[ \Rightarrow 80 = 4 n^2 - 32n\]
\[ \Rightarrow n^2 - 8n - 20 = 0\]
\[ \Rightarrow (n - 10)(n + 2) = 0\]
\[ \Rightarrow n = 10, - 2\]
\[\text { But, n cannot be negative } . \]
\[ \therefore n = 10 \]
संबंधित प्रश्न
Find the sum of odd integers from 1 to 2001.
In an A.P., if pth term is 1/q and qth term is 1/p, prove that the sum of first pq terms is 1/2 (pq + 1) where `p != q`
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.
The difference between any two consecutive interior angles of a polygon is 5°. If the smallest angle is 120°, find the number of the sides of the polygon.
Show that the sum of (m + n)th and (m – n)th terms of an A.P. is equal to twice the mth term.
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?
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
9, 7, 5, 3, ...
The 10th and 18th terms of an A.P. are 41 and 73 respectively. Find 26th term.
An A.P. consists of 60 terms. If the first and the last terms be 7 and 125 respectively, find 32nd term.
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.
If the sum of three numbers in A.P. is 24 and their product is 440, find the numbers.
Find the sum of the following arithmetic progression :
1, 3, 5, 7, ... to 12 terms
Find the sum of the following arithmetic progression :
(x − y)2, (x2 + y2), (x + y)2, ... to n terms
Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.
Solve:
1 + 4 + 7 + 10 + ... + x = 590.
Find the r th term of an A.P., the sum of whose first n terms is 3n2 + 2n.
The sum of first 7 terms of an A.P. is 10 and that of next 7 terms is 17. Find the progression.
Find the sum of n terms of the A.P. whose kth terms is 5k + 1.
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
If the sum of a certain number of terms of the AP 25, 22, 19, ... is 116. Find the last term.
If the sum of n terms of an A.P. is nP + \[\frac{1}{2}\] n (n − 1) Q, where P and Q are constants, find the common difference.
If a, b, c is in A.P., then show that:
bc − a2, ca − b2, ab − c2 are in A.P.
If \[a\left( \frac{1}{b} + \frac{1}{c} \right), b\left( \frac{1}{c} + \frac{1}{a} \right), c\left( \frac{1}{a} + \frac{1}{b} \right)\] are in A.P., prove that a, b, c are in A.P.
Show that x2 + xy + y2, z2 + zx + x2 and y2 + yz + z2 are consecutive terms of an A.P., if x, y and z are in A.P.
A manufacturer of radio sets produced 600 units in the third year and 700 units in the seventh year. Assuming that the product increases uniformly by a fixed number every year, find (i) the production in the first year (ii) the total product in 7 years and (iii) the product in the 10th year.
A piece of equipment cost a certain factory Rs 600,000. If it depreciates in value, 15% the first, 13.5% the next year, 12% the third year, and so on. What will be its value at the end of 10 years, all percentages applying to the original cost?
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?
If the sums of n terms of two arithmetic progressions are in the ratio 2n + 5 : 3n + 4, then write the ratio of their m th terms.
Write the value of n for which n th terms of the A.P.s 3, 10, 17, ... and 63, 65, 67, .... are equal.
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 the first, second and last term of an A.P are a, b and 2a respectively, then its sum is
Mark the correct alternative in the following question:
Let Sn denote the sum of first n terms of an A.P. If S2n = 3Sn, then S3n : Sn is equal to
If second, third and sixth terms of an A.P. are consecutive terms of a G.P., write the common ratio of the G.P.
Write the quadratic equation the arithmetic and geometric means of whose roots are Aand G respectively.
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 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)`
If a1, a2, a3, .......... are an A.P. such that a1 + a5 + a10 + a15 + a20 + a24 = 225, then a1 + a2 + a3 + ...... + a23 + a24 is equal to ______.
The internal angles of a convex polygon are in A.P. The smallest angle is 120° and the common difference is 5°. The number to sides of the polygon is ______.
