Advertisements
Advertisements
प्रश्न
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely real ?
Advertisements
उत्तर
12 + 8i, 11 + 6i, 10 + 4i...
This is an A.P.
Here, we have:
a = 12 + 8i
\[d = \left( 11 + 6i - 12 - 8i \right)\]
\[ = \left( - 1 - 2i \right)\]
\[\text { Let the real term be } a_n = a + \left( n - 1 \right)d . \]
\[ a_n = \left( 12 + 8i \right) + \left( n - 1 \right)\left( - 1 - 2i \right)\]
\[ = \left( 12 + 8i \right) + \left( - n + 1 - 2in + 2i \right)\]
\[ = 12 + 8i - n + 1 - 2in + 2i\]
\[ = \left( 13 - n \right) + \left( 8 - 2n + 2 \right)i\]
\[ = \left( 13 - n \right) + \left( 10 - 2n \right)i\]
\[ a_n \text { has to be real } . \]
\[ \therefore \left( 10 - 2n \right) = 0\]
\[ \Rightarrow n = 5\]
APPEARS IN
संबंधित प्रश्न
Find the sum of odd integers from 1 to 2001.
How many terms of the A.P. -6 , `-11/2` , -5... are needed to give the sum –25?
If the sum of n terms of an A.P. is 3n2 + 5n and its mth term is 164, find the value of m.
The sum of the first four terms of an A.P. is 56. The sum of the last four terms is 112. If its first term is 11, then find the number of terms.
The pth, qth and rth terms of an A.P. are a, b, c respectively. Show that (q – r )a + (r – p )b + (p – q )c = 0
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.
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, ...?
How many terms are there in the A.P. 7, 10, 13, ... 43 ?
The 4th term of an A.P. is three times the first and the 7th term exceeds twice the third term by 1. Find the first term and the common difference.
Find the second term and nth term of an A.P. whose 6th term is 12 and the 8th term is 22.
An A.P. consists of 60 terms. If the first and the last terms be 7 and 125 respectively, find 32nd term.
\[\text { If } \theta_1 , \theta_2 , \theta_3 , . . . , \theta_n \text { are in AP, whose common difference is d, then show that }\]
\[\sec \theta_1 \sec \theta_2 + \sec \theta_2 \sec \theta_3 + . . . + \sec \theta_{n - 1} \sec \theta_n = \frac{\tan \theta_n - \tan \theta_1}{\sin d} \left[ NCERT \hspace{0.167em} EXEMPLAR \right]\]
Find the sum of the following arithmetic progression :
\[\frac{x - y}{x + y}, \frac{3x - 2y}{x + y}, \frac{5x - 3y}{x + y}\], ... to n terms.
Find the sum of the following serie:
101 + 99 + 97 + ... + 47
Find the sum of all odd numbers between 100 and 200.
Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.
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 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.
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 \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P., prove that:
a (b +c), b (c + a), c (a +b) 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.
Write the common difference of an A.P. whose nth term is xn + y.
If log 2, log (2x − 1) and log (2x + 3) are in A.P., write the value of x.
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
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 =
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:
Let Sn denote the sum of first n terms of an A.P. If S2n = 3Sn, then S3n : Sn is equal to
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
If a1, a2, ..., an are in A.P. with common difference d (where d ≠ 0); then the sum of the series sin d (cosec a1 cosec a2 + cosec a2 cosec a3 + ...+ cosec an–1 cosec an) is equal to cot a1 – cot an
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
If n AM's are inserted between 1 and 31 and ratio of 7th and (n – 1)th A.M. is 5:9, then n equals ______.
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 ______.
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 ______.
