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:
\[a_{25} = 0\]
\[ \Rightarrow \text { LHS }: a_{25} = a + \left( 25 - 1 \right)d \]
\[ = a + 24d\]
\[ = - 24d + 24d \left( \text { From }(i) \right)\]
\[ = 0 = \text { RHS }\]
\[\text { Hence, proved } .\]
APPEARS IN
संबंधित प्रश्न
Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5.
The sums of n terms of two arithmetic progressions are in the ratio 5n + 4: 9n + 6. Find the ratio of their 18th terms
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)
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.
Show that the sum of (m + n)th and (m – n)th terms of an A.P. is equal to twice the mth term.
Let < an > be a sequence. Write the first five term in the following:
a1 = 1, an = an − 1 + 2, n ≥ 2
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, ...
Find:
nth term of the A.P. 13, 8, 3, −2, ...
Which term of the A.P. 3, 8, 13, ... is 248?
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely imaginary?
In a certain A.P. the 24th term is twice the 10th term. Prove that the 72nd term is twice the 34th 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.
Find the sum of the following arithmetic progression :
41, 36, 31, ... to 12 terms
Find the sum of the following serie:
2 + 5 + 8 + ... + 182
Find the sum of all natural numbers between 1 and 100, which are divisible by 2 or 5.
The sum of first 7 terms of an A.P. is 10 and that of next 7 terms is 17. Find the progression.
If the 5th and 12th terms of an A.P. are 30 and 65 respectively, what is the sum of first 20 terms?
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.
Find an A.P. in which the sum of any number of terms is always three times the squared number of these 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 \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P., prove that:
a (b +c), b (c + a), c (a +b) are in A.P.
Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.
A man saved ₹66000 in 20 years. In each succeeding year after the first year he saved ₹200 more than what he saved in the previous year. How much did he save in the first year?
Write the sum of first n odd natural numbers.
If the sum of n terms of an A.P. is 2 n2 + 5 n, then its nth term is
If a1, a2, a3, .... an are in A.P. with common difference d, then the sum of the series sin d [sec a1 sec a2 + sec a2 sec a3 + .... + sec an − 1 sec an], is
If n arithmetic means are inserted between 1 and 31 such that the ratio of the first mean and nth mean is 3 : 29, then the value of n is
Mark the correct alternative in the following question:
If in an A.P., the pth term is q and (p + q)th term is zero, then the qth term is
The first three of four given numbers are in G.P. and their last three are in A.P. with common difference 6. If first and fourth numbers are equal, then the first number is
If a, b, c are in G.P. and a1/x = b1/y = c1/z, then xyz are in
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
In an A.P. the pth term is q and the (p + q)th term is 0. Then the qth term is ______.
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 ______.
