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:
\[a_1 + a_3 + . . . + a_{2n - 1} = 24 . . . (1)\]
\[ a_2 + a_4 + . . . + a_{2n} = 30 . . . (2)\]
\[\text { Subtracting (1) from (2), we get: } \]
\[\left( d + d + . . . + \text { upto n terms } \right) = 6\]
\[ \Rightarrow nd = 6 . . . (3)\]
\[\text { Given }: \]
\[ a_{2n} = a_1 + \frac{21}{2}\]
\[ \Rightarrow a_{2n} - a_1 = \frac{21}{2}\]
\[ \Rightarrow a + (2n - 1)d - a = \frac{21}{2} [ \because a_{2n} = a + (2n - 1)d, a_1 = a]\]
\[ \Rightarrow 2nd - d = \frac{21}{2}\]
\[ \Rightarrow 2 \times 6 - d = \frac{21}{2} \left( \text { From }(3) \right)\]
\[ \Rightarrow d = \frac{3}{2}\]
\[\text { Putting the value in (3), we get: } \]
\[n = 4\]
\[ \Rightarrow 2n = 8\]
\[\text { Thus, there are 8 terms in the progression } . \]
\[\text { To find the value of the first term: } \]
\[ a_2 + a_4 + . . . + a_{2n} = 30\]
\[ \Rightarrow (a + d) + (a + 3d) + . . . + [a + (2n - 1)d] = 30\]
\[ \Rightarrow \frac{n}{2}\left[ \left( a + d \right) + a + (2n - 1)d \right] = 30\]
\[\text { Putting n = 4 and d }= \frac{3}{2}, \text { we get: } \]
\[ a = \frac{3}{2}\]
\[\text { So, the series will be } 1\frac{1}{2}, 3, 4\frac{1}{2} . . .\]
संबंधित प्रश्न
How many terms of the A.P. -6 , `-11/2` , -5... are needed to give the sum –25?
If the sum of a certain number of terms of the A.P. 25, 22, 19, … is 116. Find the last term
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)
if `(a^n + b^n)/(a^(n-1) + b^(n-1))` is the A.M. between a and b, then find the value of n.
Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A.P. and the ratio of 7th and (m – 1)th numbers is 5:9. Find the value of m.
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.
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
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.
Let < an > be a sequence. Write the first five term in the following:
a1 = 1 = a2, an = an − 1 + an − 2, n > 2
Let < an > be a sequence. Write the first five term in the following:
a1 = a2 = 2, an = an − 1 − 1, n > 2
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2}, 7\sqrt{2}, . . .\]
The 6th and 17th terms of an A.P. are 19 and 41 respectively, find the 40th term.
If 9th term of an A.P. is zero, prove that its 29th term is double the 19th term.
Find the 12th term from the following arithmetic progression:
3, 8, 13, ..., 253
\[\text { If } \theta_1 , \theta_2 , \theta_3 , . . . , \theta_n \text { are in AP, whose common difference is d, then show that }\]
\[\sec \theta_1 \sec \theta_2 + \sec \theta_2 \sec \theta_3 + . . . + \sec \theta_{n - 1} \sec \theta_n = \frac{\tan \theta_n - \tan \theta_1}{\sin d} \left[ NCERT \hspace{0.167em} EXEMPLAR \right]\]
Find the sum of all natural numbers between 1 and 100, which are divisible by 2 or 5.
Find the sum of all odd numbers between 100 and 200.
Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.
The third term of an A.P. is 7 and the seventh term exceeds three times the third term by 2. Find the first term, the common difference and the sum of first 20 terms.
If the sum of a certain number of terms of the AP 25, 22, 19, ... is 116. Find the last term.
Find an A.P. in which the sum of any number of terms is always three times the squared number of these terms.
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 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 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.
A man accepts a position with an initial salary of ₹5200 per month. It is understood that he will receive an automatic increase of ₹320 in the very next month and each month thereafter.
(i) Find his salary for the tenth month.
(ii) What is his total earnings during the first year?
Write the common difference of an A.P. the sum of whose first n terms is
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 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 for an arithmetic progression, 9 times nineth term is equal to 13 times thirteenth term, then value of twenty second term is ____________.
If a, b, c, d are four distinct positive quantities in A.P., then show that bc > ad
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. What is his total earnings during the first year?
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 ______.
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
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 ______.
