Advertisements
Advertisements
प्रश्न
The number of terms of an A.P. is even; the sum of odd terms is 24, of the even terms is 30, and the last term exceeds the first by \[10 \frac{1}{2}\] , find the number of terms and the series.
Advertisements
उत्तर
Let total number of terms be 2n.
According to question, we have:
\[a_1 + a_3 + . . . + a_{2n - 1} = 24 . . . (1)\]
\[ a_2 + a_4 + . . . + a_{2n} = 30 . . . (2)\]
\[\text { Subtracting (1) from (2), we get: } \]
\[\left( d + d + . . . + \text { upto n terms } \right) = 6\]
\[ \Rightarrow nd = 6 . . . (3)\]
\[\text { Given }: \]
\[ a_{2n} = a_1 + \frac{21}{2}\]
\[ \Rightarrow a_{2n} - a_1 = \frac{21}{2}\]
\[ \Rightarrow a + (2n - 1)d - a = \frac{21}{2} [ \because a_{2n} = a + (2n - 1)d, a_1 = a]\]
\[ \Rightarrow 2nd - d = \frac{21}{2}\]
\[ \Rightarrow 2 \times 6 - d = \frac{21}{2} \left( \text { From }(3) \right)\]
\[ \Rightarrow d = \frac{3}{2}\]
\[\text { Putting the value in (3), we get: } \]
\[n = 4\]
\[ \Rightarrow 2n = 8\]
\[\text { Thus, there are 8 terms in the progression } . \]
\[\text { To find the value of the first term: } \]
\[ a_2 + a_4 + . . . + a_{2n} = 30\]
\[ \Rightarrow (a + d) + (a + 3d) + . . . + [a + (2n - 1)d] = 30\]
\[ \Rightarrow \frac{n}{2}\left[ \left( a + d \right) + a + (2n - 1)d \right] = 30\]
\[\text { Putting n = 4 and d }= \frac{3}{2}, \text { we get: } \]
\[ a = \frac{3}{2}\]
\[\text { So, the series will be } 1\frac{1}{2}, 3, 4\frac{1}{2} . . .\]
संबंधित प्रश्न
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.
if `(a^n + b^n)/(a^(n-1) + b^(n-1))` is the A.M. between a and b, then find the value of n.
If the sum of three numbers in A.P., is 24 and their product is 440, find the numbers.
A person writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four different persons with instruction that they move the chain similarly. Assuming that the chain is not broken and that it costs 50 paise to mail one letter. Find the amount spent on the postage when 8th set of letter is mailed.
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 = a2 = 2, an = an − 1 − 1, 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.
If the sequence < an > is an A.P., show that am +n +am − n = 2am.
Which term of the A.P. 84, 80, 76,... is 0?
Is 68 a term of the A.P. 7, 10, 13, ...?
The 6th and 17th terms of an A.P. are 19 and 41 respectively, find the 40th 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.
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 second term and nth term of an A.P. whose 6th term is 12 and the 8th term is 22.
Find the sum of first n natural numbers.
Find the sum of all integers between 100 and 550, which are divisible by 9.
The sum of first 7 terms of an A.P. is 10 and that of next 7 terms is 17. Find the progression.
If Sn = n2 p and Sm = m2 p, m ≠ n, in an A.P., prove that Sp = p3.
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
How many terms of the A.P. −6, \[- \frac{11}{2}\], −5, ... are needed to give the sum −25?
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:
\[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] 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:
a2 + c2 + 4ac = 2 (ab + bc + ca)
If a, b, c is in A.P., prove that:
a3 + c3 + 6abc = 8b3.
A man saves Rs 32 during the first year. Rs 36 in the second year and in this way he increases his savings by Rs 4 every year. Find in what time his saving will be Rs 200.
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?
A man accepts a position with an initial salary of ₹5200 per month. It is understood that he will receive an automatic increase of ₹320 in the very next month and each month thereafter.
(i) Find his salary for the tenth month.
(ii) What is his total earnings during the first year?
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
If four numbers in A.P. are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are
If, S1 is the sum of an arithmetic progression of 'n' odd number of terms and S2 the sum of the terms of the series in odd places, then \[\frac{S_1}{S_2}\] =
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
The first three of four given numbers are in G.P. and their last three are in A.P. with common difference 6. If first and fourth numbers are equal, then the first number is
If there are (2n + 1) terms in an A.P., then prove that the ratio of the sum of odd terms and the sum of even terms is (n + 1) : n
A man saved Rs 66000 in 20 years. In each succeeding year after the first year he saved Rs 200 more than what he saved in the previous year. How much did he save in the first year?
If in an A.P., Sn = qn2 and Sm = qm2, where Sr denotes the sum of r terms of the A.P., then Sq equals ______.
If b2, a2, c2 are in A.P., then `1/(a + b), 1/(b + c), 1/(c + a)` will be in ______
What is the general form of an A.P. with first term \(a\) and common difference \(d\)?
Which of the following is an example of a Constant A.P.?
