Advertisements
Advertisements
प्रश्न
The sum of terms equidistant from the beginning and end in an A.P. is equal to ______.
उत्तर
The sum of terms equidistant from the beginning and end in an A.P. is equal to the [first term + last term].
Explanation:
Let A.P be a, a + d, a + 2d, a + 3d, …, a + (n – 1)d
Taking first and last term
a1 + an = a + a + (n – 1)d
= 2a + (n – 1)d
Taking second and second last term
a2 + an–1 = (a + d) + [a + (n – 2)d]
= 2a + (n – 1)d = a1 + an
Taking third from the beginning and the third from the end
a3 + an–2 = (a + 2d) + [a + (n – 3)d]
= 2a + (n – 1)d
= a1 + an
From the above pattern, we observe that the sum of terms equidistant from the beginning and the end in an A.P is equal to the [first term + last term]
APPEARS IN
संबंधित प्रश्न
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.
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.
Let < an > be a sequence. Write the first five term in the following:
a1 = 1, an = an − 1 + 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
Find:
10th term of the A.P. 1, 4, 7, 10, ...
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely real ?
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.
Find the 12th term from the following arithmetic progression:
1, 4, 7, 10, ..., 88
The first and the last terms of an A.P. are a and l respectively. Show that the sum of nthterm from the beginning and nth term from the end is a + l.
Find the sum of the following arithmetic progression :
(x − y)2, (x2 + y2), (x + y)2, ... to n terms
Find the sum of first n natural numbers.
Find the sum of all integers between 50 and 500 which are divisible by 7.
Find the sum of all integers between 100 and 550, which are divisible by 9.
Solve:
25 + 22 + 19 + 16 + ... + x = 115
How many terms are there in the A.P. whose first and fifth terms are −14 and 2 respectively and the sum of the terms is 40?
Find the sum of n terms of the A.P. whose kth terms is 5k + 1.
If a2, b2, c2 are in A.P., prove that \[\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}\] are in A.P.
A man arranges to pay off a debt of Rs 3600 by 40 annual instalments which form an arithmetic series. When 30 of the instalments are paid, he dies leaving one-third of the debt unpaid, find the value of the first 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.
Write the sum of first n odd natural numbers.
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 terms of an A.P. are 1 and 11. If the sum of its terms is 36, then the number of terms will be
If a1, a2, a3, .... an are in A.P. with common difference d, then the sum of the series sin d [cosec a1cosec a2 + cosec a1 cosec a3 + .... + cosec an − 1 cosec an] 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
Mark the correct alternative in the following question:
\[\text { If in an A . P } . S_n = n^2 q \text { and } S_m = m^2 q, \text { where } S_r \text{ denotes the sum of r terms of the A . P . , then }S_q \text { equals }\]
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 the sum of n terms of a sequence is quadratic expression then it always represents an A.P
