Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Let total number of terms be 2n.
According to question, we have:
\[a_1 + a_3 + . . . + a_{2n - 1} = 24 . . . (1)\]
\[ a_2 + a_4 + . . . + a_{2n} = 30 . . . (2)\]
\[\text { Subtracting (1) from (2), we get: } \]
\[\left( d + d + . . . + \text { upto n terms } \right) = 6\]
\[ \Rightarrow nd = 6 . . . (3)\]
\[\text { Given }: \]
\[ a_{2n} = a_1 + \frac{21}{2}\]
\[ \Rightarrow a_{2n} - a_1 = \frac{21}{2}\]
\[ \Rightarrow a + (2n - 1)d - a = \frac{21}{2} [ \because a_{2n} = a + (2n - 1)d, a_1 = a]\]
\[ \Rightarrow 2nd - d = \frac{21}{2}\]
\[ \Rightarrow 2 \times 6 - d = \frac{21}{2} \left( \text { From }(3) \right)\]
\[ \Rightarrow d = \frac{3}{2}\]
\[\text { Putting the value in (3), we get: } \]
\[n = 4\]
\[ \Rightarrow 2n = 8\]
\[\text { Thus, there are 8 terms in the progression } . \]
\[\text { To find the value of the first term: } \]
\[ a_2 + a_4 + . . . + a_{2n} = 30\]
\[ \Rightarrow (a + d) + (a + 3d) + . . . + [a + (2n - 1)d] = 30\]
\[ \Rightarrow \frac{n}{2}\left[ \left( a + d \right) + a + (2n - 1)d \right] = 30\]
\[\text { Putting n = 4 and d }= \frac{3}{2}, \text { we get: } \]
\[ a = \frac{3}{2}\]
\[\text { So, the series will be } 1\frac{1}{2}, 3, 4\frac{1}{2} . . .\]
APPEARS IN
संबंधित प्रश्न
Find the sum of odd integers from 1 to 2001.
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`
Find the sum to n terms of the A.P., whose kth term is 5k + 1.
If the sum of n terms of an A.P. is 3n2 + 5n and its mth term is 164, find the value of m.
Show that the sum of (m + n)th and (m – n)th terms of an A.P. is equal to twice the mth term.
Is 302 a term of the A.P. 3, 8, 13, ...?
The 10th and 18th terms of an A.P. are 41 and 73 respectively. Find 26th term.
The 4th term of an A.P. is three times the first and the 7th term exceeds twice the third term by 1. Find the first term and the common difference.
\[\text { If } \theta_1 , \theta_2 , \theta_3 , . . . , \theta_n \text { are in AP, whose common difference is d, then show that }\]
\[\sec \theta_1 \sec \theta_2 + \sec \theta_2 \sec \theta_3 + . . . + \sec \theta_{n - 1} \sec \theta_n = \frac{\tan \theta_n - \tan \theta_1}{\sin d} \left[ NCERT \hspace{0.167em} EXEMPLAR \right]\]
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.
The angles of a quadrilateral are in A.P. whose common difference is 10°. Find the angles.
Find the sum of the following arithmetic progression :
41, 36, 31, ... to 12 terms
Find the sum of the following arithmetic progression :
a + b, a − b, a − 3b, ... to 22 terms
Solve:
25 + 22 + 19 + 16 + ... + x = 115
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.
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., then show that:
bc − a2, ca − b2, ab − c2 are in A.P.
If a, b, c is in A.P., prove that:
(a − c)2 = 4 (a − b) (b − c)
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.
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 man arranges to pay off a debt of Rs 3600 by 40 annual instalments which form an arithmetic series. When 30 of the instalments are paid, he dies leaving one-third of the debt unpaid, find the value of the first instalment.
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 carpenter was hired to build 192 window frames. The first day he made five frames and each day thereafter he made two more frames than he made the day before. How many days did it take him to finish the job?
If the sum of n terms of an AP is 2n2 + 3n, then write its nth term.
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 \[\frac{3 + 5 + 7 + . . . + \text { upto n terms }}{5 + 8 + 11 + . . . . \text { upto 10 terms }}\] 7, then find the value of n.
Mark the correct alternative in the following question:
The 10th common term between the A.P.s 3, 7, 11, 15, ... and 1, 6, 11, 16, ... is
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.
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
If a, b, c are in G.P. and a1/x = b1/y = c1/z, then xyz are in
If for an arithmetic progression, 9 times nineth term is equal to 13 times thirteenth term, then value of twenty second term is ____________.
If there are (2n + 1) terms in an A.P., then prove that the ratio of the sum of odd terms and the sum of even terms is (n + 1) : n
Any term of an A.P. (except first) is equal to half the sum of terms which are equidistant from it.
If the sum of n terms of a sequence is quadratic expression then it always represents an A.P
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 ______.
