Advertisements
Advertisements
Question
If the first, second and last term of an A.P are a, b and 2a respectively, then its sum is
Options
\[\frac{ab}{2 (b - a)}\]
\[\frac{ab}{b - a}\]
\[\frac{3 ab}{2 (b - a)}\]
none of these
Advertisements
Solution
\[\frac{3 ab}{2 (b - a)}\]
Let the A.P. be a, a+d, a+2d........a+nd.
Here, let d be the common difference and n be the total number of terms.
\[a_1 = a, \]
\[ a_2 = b\]
\[ \Rightarrow a + d = b\]
\[ \Rightarrow d = b - a . . . . . \left( 1 \right)\]
\[ a_n = 2a\]
\[ \Rightarrow a + \left( n - 1 \right)d = 2a\]
\[ \Rightarrow \left( n - 1 \right)d = a\]
\[ \Rightarrow d = \frac{a}{n - 1} . . . . . \left( 2 \right)\]
Given:
From equations \[\left( 1 \right) \text { and } \left( 2 \right),\] we have:
\[\Rightarrow \frac{a}{n - 1} = b - a\]
\[ \Rightarrow \frac{a}{b - a} + 1 = n\]
\[ \Rightarrow \frac{a + b - a}{b - a} = n\]
\[ \Rightarrow \frac{b}{b - a} = n\]
Now, sum of n terms of an A.P.:
\[S = \frac{n}{2}\left\{ a + a_n \right\}\]
\[ = \frac{n}{2}\left( 3a \right)\]
\[ = \frac{3ab}{2\left( b - a \right)}\]
APPEARS IN
RELATED QUESTIONS
Find the sum of odd integers from 1 to 2001.
Sum of the first p, q and r terms of an A.P. are a, b and c, respectively.
Prove that `a/p (q - r) + b/q (r- p) + c/r (p - q) = 0`
Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.
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.
Let < an > be a sequence. Write the first five term in the following:
a1 = 1 = a2, an = an − 1 + an − 2, n > 2
The Fibonacci sequence is defined by a1 = 1 = a2, an = an − 1 + an − 2 for n > 2
Find `(""^an +1)/(""^an")` for n = 1, 2, 3, 4, 5.
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
−1, 1/4, 3/2, 11/4, ...
If the sequence < an > is an A.P., show that am +n +am − n = 2am.
Which term of the sequence 24, \[23\frac{1}{4,} 22\frac{1}{2,} 21\frac{3}{4}\]....... is the first negative term?
Find the second term and nth term of an A.P. whose 6th term is 12 and the 8th term is 22.
Find the four numbers in A.P., whose sum is 50 and in which the greatest number is 4 times the least.
Find the sum of the following arithmetic progression :
1, 3, 5, 7, ... to 12 terms
Find the sum of the following arithmetic progression :
(x − y)2, (x2 + y2), (x + y)2, ... to n terms
Find the sum of all natural numbers between 1 and 100, which are divisible by 2 or 5.
Find the sum of all odd numbers between 100 and 200.
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 the 5th and 12th terms of an A.P. are 30 and 65 respectively, what is the sum of first 20 terms?
Find the sum of odd integers from 1 to 2001.
If the sum of n terms of an A.P. is nP + \[\frac{1}{2}\] n (n − 1) Q, where P and Q are constants, find the common difference.
If a, b, c is in A.P., then show that:
bc − a2, ca − b2, ab − c2 are in A.P.
If \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:
bc, ca, ab are in A.P.
If a, b, c is in A.P., prove that:
a2 + c2 + 4ac = 2 (ab + bc + ca)
Show that x2 + xy + y2, z2 + zx + x2 and y2 + yz + z2 are consecutive terms of an A.P., if x, y and z are in A.P.
Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.
Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in annual instalments of Rs 1000 plus 10% interest on the unpaid amount. How much the scooter will cost him.
Write the common difference of an A.P. whose nth term is xn + y.
Write the common difference of an A.P. the sum of whose first n terms is
Write the sum of first n even natural numbers.
Write the value of n for which n th terms of the A.P.s 3, 10, 17, ... and 63, 65, 67, .... are equal.
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. be 3 n2 − n and its common difference is 6, then its first term 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
Mark the correct alternative in the following question:
\[\text { If in an A . P } . S_n = n^2 q \text { and } S_m = m^2 q, \text { where } S_r \text{ denotes the sum of r terms of the A . P . , then }S_q \text { equals }\]
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
The sum of n terms of an AP is 3n2 + 5n. The number of term which equals 164 is ______.
The number of terms in an A.P. is even; the sum of the odd terms in lt is 24 and that the even terms is 30. If the last term exceeds the first term by `10 1/2`, then the number of terms in the A.P. is ______.
