Advertisements
Advertisements
प्रश्न
If 9th term of an A.P. is zero, prove that its 29th term is double the 19th term.
Advertisements
उत्तर
Given:
\[a_9 = 0 \]
\[ \Rightarrow a + \left( 9 - 1 \right)d = 0 \left[ a_n = a + \left( n - 1 \right)d \right]\]
\[ \Rightarrow a + 8d = 0\]
\[ \Rightarrow a = - 8d . . . (i)\]
To prove:
\[a_{29} = 2 a_{19} \]
Proof:
\[\text { LHS }: a_{29} = a + \left( 29 - 1 \right)d\]
\[ = a + 28d\]
\[ = - 8d + 28d \left( \text { From }(i) \right)\]
\[ = 20d\]
\[RHS: 2 a_{19} = 2\left[ a + \left( 19 - 1 \right)d \right]\]
\[ = 2(a + 18d)\]
\[ = 2a + 36d\]
\[ = 2( - 8d) + 36d \left( \text { From }(i) \right)\]
\[ = - 16d + 36d\]
\[ = 20d\]
LHS = RHS Hence, proved.
APPEARS IN
संबंधित प्रश्न
How many terms of the A.P. -6 , `-11/2` , -5... are needed to give the sum –25?
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?
Show that the sum of (m + n)th and (m – n)th terms of an A.P. is equal to twice the mth term.
If the sum of three numbers in A.P., is 24 and their product is 440, find the numbers.
A man deposited Rs 10000 in a bank at the rate of 5% simple interest annually. Find the amount in 15th year since he deposited the amount and also calculate the total amount after 20 years.
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.
\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2}, 7\sqrt{2}, . . .\]
The nth term of a sequence is given by an = 2n + 7. Show that it is an A.P. Also, find its 7th term.
Which term of the A.P. 4, 9, 14, ... is 254?
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely imaginary?
How many terms are there in the A.P. 7, 10, 13, ... 43 ?
If the nth term of the A.P. 9, 7, 5, ... is same as the nth term of the A.P. 15, 12, 9, ... find n.
How many numbers are there between 1 and 1000 which when divided by 7 leave remainder 4?
Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.
Solve:
1 + 4 + 7 + 10 + ... + x = 590.
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 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:
a2 + c2 + 4ac = 2 (ab + bc + ca)
A man saves Rs 32 during the first year. Rs 36 in the second year and in this way he increases his savings by Rs 4 every year. Find in what time his saving will be Rs 200.
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?
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
In n A.M.'s are introduced between 3 and 17 such that the ratio of the last mean to the first mean is 3 : 1, then the value of n is
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 =
If the sum of first n even natural numbers is equal to k times the sum of first n odd natural numbers, then k =
If in an A.P., Sn = n2p and Sm = m2p, where Sr denotes the sum of r terms of the A.P., then Sp is equal to
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
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
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
The first term of an A.P. is a, the second term is b and the last term is c. Show that the sum of the A.P. is `((b + c - 2a)(c + a))/(2(b - a))`.
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
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.
The product of three numbers in A.P. is 224, and the largest number is 7 times the smallest. Find the numbers
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 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 ______.
The sum of terms equidistant from the beginning and end in an A.P. is equal to ______.
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 ______.
The sum of n terms of an AP is 3n2 + 5n. The number of term which equals 164 is ______.
The internal angles of a convex polygon are in A.P. The smallest angle is 120° and the common difference is 5°. The number to sides of the polygon is ______.
