Advertisements
Advertisements
प्रश्न
If the sum of n terms of a sequence is quadratic expression then it always represents an A.P
पर्याय
True
False
उत्तर
This statement is False.
Explanation:
Let Sn = an2 + bn + c ...(Quadratic expression)
S1 = a + b + c
∴ a1 = a + b + c
S2 = 4a + 2b + c
a2 = S2 – S1
= (4a + 2b + c) – (a + b + c)
= 3a + b
S3 = 9a + 3b + c
⇒ a3 = S3 – S2
= (9a + 3b + c) – (4a + 2b + c)
= 5a + b
Common difference d = a2 – a1
= (3a + b) – (a + b + c)
= 2a – c
and d = a3 – a2 = (5a + b) – (3a + b) = 2a
Here, we observe that a2 – a1 ≠ a3 – a2
So it does not represent an A.P
APPEARS IN
संबंधित प्रश्न
In an A.P., if pth term is 1/q and qth term is 1/p, prove that the sum of first pq terms is 1/2 (pq + 1) where `p != q`
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.
Let the sum of n, 2n, 3n terms of an A.P. be S1, S2 and S3, respectively, show that S3 = 3 (S2– S1)
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, ...
If the sequence < an > is an A.P., show that am +n +am − n = 2am.
How many terms are there in the A.P. 7, 10, 13, ... 43 ?
How many terms are there in the A.P.\[- 1, - \frac{5}{6}, -\frac{2}{3}, - \frac{1}{2}, . . . , \frac{10}{3}?\]
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.
If 9th term of an A.P. is zero, prove that its 29th term is double the 19th term.
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.
Find the sum of the following arithmetic progression :
(x − y)2, (x2 + y2), (x + y)2, ... to n terms
Find the sum of all integers between 100 and 550, which are divisible by 9.
Find the sum of odd integers from 1 to 2001.
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 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:
a3 + c3 + 6abc = 8b3.
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.
The income of a person is Rs 300,000 in the first year and he receives an increase of Rs 10000 to his income per year for the next 19 years. Find the total amount, he received in 20 years.
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?
If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is
If the sum of n terms of an A.P., is 3 n2 + 5 n then which of its terms is 164?
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 first, second and last term of an A.P are a, b and 2a respectively, then its sum is
If second, third and sixth terms of an A.P. are consecutive terms of a G.P., write the common ratio of the G.P.
The product of three numbers in A.P. is 224, and the largest number is 7 times the smallest. Find the numbers
Any term of an A.P. (except first) is equal to half the sum of terms which are equidistant from it.
The sum of n terms of an AP is 3n2 + 5n. The number of term which equals 164 is ______.
If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A.P. is ______.
If 100 times the 100th term of an A.P. with non zero common difference equals the 50 times its 50th term, then the 150th term of this A.P. is ______.
