Advertisements
Advertisements
प्रश्न
Solve:
25 + 22 + 19 + 16 + ... + x = 115
Advertisements
उत्तर
25 + 22 + 19 + 16 + ... + x = 115
Here, a = 25, d = \[-\] 3,
\[S_n = 115\]
\[\text { We know }: \]
\[ S_n = \frac{n}{2}\left[ 2a + (n - 1)d \right]\]
\[ \Rightarrow 115 = \frac{n}{2}\left[ 2 \times 25 + (n - 1) \times \left( - 3 \right) \right]\]
\[ \Rightarrow 115 \times 2 = n\left[ 50 - 3n + 3 \right]\]
\[ \Rightarrow 230 = n\left( 53 - 3n \right)\]
\[ \Rightarrow 230 = 53n - 3 n^2 \]
\[ \Rightarrow 3 n^2 - 53n + 230 = 0\]
\[\text { By quadratic formula: } \]
\[ n = \frac{- b \pm \sqrt{b^2 - 4ac}}{2a}\]
\[\text { Substituting a = 3, b = - 53 and c = 230, we get: } \]
\[n = \frac{53 \pm \sqrt{\left( 53 \right)^2 - 4 \times 3 \times 230}}{2 \times 3} = \frac{46}{6}, 10\]
\[ \Rightarrow n = 10, as n \neq \frac{46}{6}\]
\[ \therefore a_n = a + (n - 1)d\]
\[ \Rightarrow x = 25 + (10 - 1)( - 3)\]
\[ \Rightarrow x = 25 - 27 = - 2\]
APPEARS IN
संबंधित प्रश्न
In an A.P, the first term is 2 and the sum of the first five terms is one-fourth of the next five terms. Show that 20th term is –112.
How many terms of the A.P. -6 , `-11/2` , -5... are needed to give the sum –25?
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 ratio of the sums of m and n terms of an A.P. is m2: n2. Show that the ratio of mth and nthterm is (2m – 1): (2n – 1)
A man starts repaying a loan as first installment of Rs. 100. If he increases the installment by Rs 5 every month, what amount he will pay in the 30th installment?
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.
Find the sum of all numbers between 200 and 400 which are divisible by 7.
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.
if `a(1/b + 1/c), b(1/c+1/a), c(1/a+1/b)` are in A.P., prove that a, b, c are in A.P.
A sequence is defined by an = n3 − 6n2 + 11n − 6, n ϵ N. Show that the first three terms of the sequence are zero and all other terms are positive.
Let < an > be a sequence. Write the first five term in the following:
a1 = 1 = a2, an = an − 1 + an − 2, n > 2
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
3, −1, −5, −9 ...
Find:
18th term of the A.P.
\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2},\]
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely imaginary?
\[\text { If } \theta_1 , \theta_2 , \theta_3 , . . . , \theta_n \text { are in AP, whose common difference is d, then show that }\]
\[\sec \theta_1 \sec \theta_2 + \sec \theta_2 \sec \theta_3 + . . . + \sec \theta_{n - 1} \sec \theta_n = \frac{\tan \theta_n - \tan \theta_1}{\sin d} \left[ NCERT \hspace{0.167em} EXEMPLAR \right]\]
Find the four numbers in A.P., whose sum is 50 and in which the greatest number is 4 times the least.
Find the sum of the series:
3 + 5 + 7 + 6 + 9 + 12 + 9 + 13 + 17 + ... to 3n terms.
Find the sum of n terms of the A.P. whose kth terms is 5k + 1.
The sums of first n terms of two A.P.'s are in the ratio (7n + 2) : (n + 4). Find the ratio of their 5th terms.
If \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P., prove that:
\[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P.
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.
If \[\frac{3 + 5 + 7 + . . . + \text { upto n terms }}{5 + 8 + 11 + . . . . \text { upto 10 terms }}\] 7, then find the value of n.
If m th term of an A.P. is n and nth term is m, then write its pth term.
If the sum of p terms of an A.P. is q and the sum of q terms is p, then the sum of p + q terms will be
If Sn denotes the sum of first n terms of an A.P. < an > such that
If the sum of n terms of an A.P., is 3 n2 + 5 n then which of its terms is 164?
If a, b, c are in A.P. and x, y, z are in G.P., then the value of xb − c yc − a za − b is
Find the sum of first 24 terms of the A.P. a1, a2, a3, ... if it is known that a1 + a5 + a10 + a15 + a20 + a24 = 225.
Show that (x2 + xy + y2), (z2 + xz + x2) and (y2 + yz + z2) are consecutive terms of an A.P., if x, y and z are in A.P.
If a1, a2, ..., an are in A.P. with common difference d (where d ≠ 0); then the sum of the series sin d (cosec a1 cosec a2 + cosec a2 cosec a3 + ...+ cosec an–1 cosec an) is equal to cot a1 – cot an
If a, b, c, d are four distinct positive quantities in A.P., then show that bc > ad
In an A.P. the pth term is q and the (p + q)th term is 0. Then the qth term 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 ______.
If the sum of n terms of a sequence is quadratic expression then it always represents an A.P
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 ______.
If n AM's are inserted between 1 and 31 and ratio of 7th and (n – 1)th A.M. is 5:9, then n equals ______.
The number of terms in an A.P. is even; the sum of the odd terms in lt is 24 and that the even terms is 30. If the last term exceeds the first term by `10 1/2`, then the number of terms in the A.P. is ______.
