Advertisements
Advertisements
Question
Any term of an A.P. (except first) is equal to half the sum of terms which are equidistant from it.
Options
True
False
Advertisements
Solution
This statement is True.
Explanation:
Let us consider an A.P a, a + d, a + 2d, …
∴ a2 + a4 = a + d + a + 3d
= 2a + 4d
= 2a3
⇒ a3 = `(a_2 + a_4)/2`
`(a_3 + a_5)/2 = (a + 2d + a + 4d)/2`
= `(2a + 6d)/2`
⇒ a + 3d
= a4
APPEARS IN
RELATED QUESTIONS
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)
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.
A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual installments of Rs 500 plus 12% interest on the unpaid amount. How much will be the tractor cost him?
A manufacturer reckons that the value of a machine, which costs him Rs 15625, will depreciate each year by 20%. Find the estimated value at the end of 5 years.
The nth term of a sequence is given by an = 2n + 7. Show that it is an A.P. Also, find its 7th term.
Is 302 a term of the A.P. 3, 8, 13, ...?
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely imaginary?
If 9th term of an A.P. is zero, prove that its 29th term is double the 19th term.
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.
In a certain A.P. the 24th term is twice the 10th term. Prove that the 72nd term is twice the 34th term.
Find the 12th term from the following arithmetic progression:
3, 5, 7, 9, ... 201
Find the 12th term from the following arithmetic progression:
1, 4, 7, 10, ..., 88
How many numbers of two digit are divisible by 3?
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 sum of three numbers in A.P. is 12 and the sum of their cubes is 288. 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 :
1, 3, 5, 7, ... to 12 terms
Find the sum of first n odd natural numbers.
Find the sum of all even integers between 101 and 999.
Find the sum of the series:
3 + 5 + 7 + 6 + 9 + 12 + 9 + 13 + 17 + ... to 3n terms.
Solve:
1 + 4 + 7 + 10 + ... + x = 590.
The first term of an A.P. is 2 and the last term is 50. The sum of all these terms is 442. Find the common difference.
If 12th term of an A.P. is −13 and the sum of the first four terms is 24, what is the sum of first 10 terms?
If the sum of a certain number of terms of the AP 25, 22, 19, ... is 116. Find the last term.
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.
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 a, b, c is in A.P., then show that:
a2 (b + c), b2 (c + a), c2 (a + b) are also in A.P.
If a, b, c is in A.P., prove that:
a3 + c3 + 6abc = 8b3.
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.
There are 25 trees at equal distances of 5 metres in a line with a well, the distance of the well from the nearest tree being 10 metres. A gardener waters all the trees separately starting from the well and he returns to the well after watering each tree to get water for the next. Find the total distance the gardener will cover in order to water all the trees.
If the sums of n terms of two arithmetic progressions are in the ratio 2n + 5 : 3n + 4, then write the ratio of their m th terms.
In the arithmetic progression whose common difference is non-zero, the sum of first 3 n terms is equal to the sum of next n terms. Then the ratio of the sum of the first 2 n terms to the next 2 nterms is
If the sum of first n even natural numbers is equal to k times the sum of first n odd natural numbers, then k =
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
Write the quadratic equation the arithmetic and geometric means of whose roots are Aand G respectively.
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
Show that (x2 + xy + y2), (z2 + xz + x2) and (y2 + yz + z2) are consecutive terms of an A.P., if x, y and z are in A.P.
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)`
