Advertisements
Advertisements
प्रश्न
If 10 times the 10th term of an A.P. is equal to 15 times the 15th term, show that 25th term of the A.P. is zero.
Advertisements
उत्तर
Given:
\[10 a_{10} = 15 a_{15} \]
\[ \Rightarrow 10\left[ a + \left( 10 - 1 \right)d \right] = 15\left[ a + \left( 15 - 1 \right)d \right]\]
\[ \Rightarrow 10(a + 9d) = 15(a + 14d)\]
\[ \Rightarrow 10a + 90d = 15a + 210d\]
\[ \Rightarrow 0 = 5a + 120d\]
\[ \Rightarrow 0 = a + 24d\]
\[ \Rightarrow a = - 24d . . . (i)\]
To show:
\[a_{25} = 0\]
\[ \Rightarrow \text { LHS }: a_{25} = a + \left( 25 - 1 \right)d \]
\[ = a + 24d\]
\[ = - 24d + 24d \left( \text { From }(i) \right)\]
\[ = 0 = \text { RHS }\]
\[\text { Hence, proved } .\]
संबंधित प्रश्न
Find the sum of odd integers from 1 to 2001.
Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5.
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)
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.
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.
−1, 1/4, 3/2, 11/4, ...
Find:
10th term of the A.P. 1, 4, 7, 10, ...
The first term of an A.P. is 5, the common difference is 3 and the last term is 80; find the number of terms.
Find the 12th term from the following arithmetic progression:
3, 8, 13, ..., 253
If < an > is an A.P. such that \[\frac{a_4}{a_7} = \frac{2}{3}, \text { find }\frac{a_6}{a_8}\].
\[\text { If } \theta_1 , \theta_2 , \theta_3 , . . . , \theta_n \text { are in AP, whose common difference is d, then show that }\]
\[\sec \theta_1 \sec \theta_2 + \sec \theta_2 \sec \theta_3 + . . . + \sec \theta_{n - 1} \sec \theta_n = \frac{\tan \theta_n - \tan \theta_1}{\sin d} \left[ NCERT \hspace{0.167em} EXEMPLAR \right]\]
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.
Three numbers are in A.P. If the sum of these numbers be 27 and the product 648, find the numbers.
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 :
50, 46, 42, ... to 10 terms
Find the sum of the following arithmetic progression :
(x − y)2, (x2 + y2), (x + y)2, ... to n terms
Find the sum of the following arithmetic progression :
\[\frac{x - y}{x + y}, \frac{3x - 2y}{x + y}, \frac{5x - 3y}{x + y}\], ... to n terms.
Find the sum of all integers between 100 and 550, which are divisible by 9.
Find the r th term of an A.P., the sum of whose first n terms is 3n2 + 2n.
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.
If S1 be the sum of (2n + 1) terms of an A.P. and S2 be the sum of its odd terms, then prove that: S1 : S2 = (2n + 1) : (n + 1).
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 \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:
\[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
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?
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 starts repaying a loan as first instalment of Rs 100 = 00. If he increases the instalments by Rs 5 every month, what amount he will pay in the 30th instalment?
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.
If the sum of n terms of an A.P. be 3 n2 − n and its common difference is 6, then its first term is
If the sum of n terms of an A.P., is 3 n2 + 5 n then which of its terms is 164?
The first and last term of an A.P. are a and l respectively. If S is the sum of all the terms of the A.P. and the common difference is given by \[\frac{l^2 - a^2}{k - (l + a)}\] , then k =
The product of three numbers in A.P. is 224, and the largest number is 7 times the smallest. Find the numbers
If the sum of m terms of an A.P. is equal to the sum of either the next n terms or the next p terms, then prove that `(m + n) (1/m - 1/p) = (m + p) (1/m - 1/n)`
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
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 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 sum of n terms of an AP is 3n2 + 5n. The number of term which equals 164 is ______.
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 ______.
