Advertisements
Advertisements
प्रश्न
The sum of first 7 terms of an A.P. is 10 and that of next 7 terms is 17. Find the progression.
Advertisements
उत्तर
\[\text { We have: } \]
\[ S_7 = 10\]
\[ \Rightarrow \frac{7}{2}\left[ 2a + (7 - 1)d \right] = 10\]
\[ \Rightarrow \frac{7}{2}\left[ 2a + 6d \right] = 10\]
\[ \Rightarrow a + 3d = \frac{10}{7} . . . (i)\]
\[\text { Also, the sum of the next seven terms } = S_{14} - S_7 = 17\]
\[ \Rightarrow \frac{14}{2}\left[ 2a + \left( 14 - 1 \right)d \right] - \frac{7}{2}\left[ 2a + (7 - 1)d \right] = 17\]
\[ \Rightarrow 7\left[ 2a + 13d \right]\]
\[ - \frac{7}{2}\left[ 2a + 6d \right] = 17\]
\[ \Rightarrow 14a + 91d - 7a - 21d = 17\]
\[ \Rightarrow 7a + 70d = 17\]
\[ \Rightarrow a + 10d = \frac{17}{7} . . . (ii)\]
\[\text { From (i) and (ii), we get }: \]
\[\frac{10}{7} - 3d = \frac{17}{7} - 10d\]
\[ \Rightarrow 7d = 1\]
\[ \Rightarrow d = \frac{1}{7}\]
\[\text { Putting the value in (i), we get: } \]
\[a + 3d = \frac{10}{7}\]
\[ \Rightarrow a + \frac{3}{7} = \frac{10}{7}\]
\[ \Rightarrow a = 1\]
\[ \therefore a = 1, d = \frac{1}{7}\]
The progression thus formed is
\[1, \frac{8}{7}, \frac{9}{7}, \frac{10}{7} . . .\]
संबंधित प्रश्न
Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5.
Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A.P. and the ratio of 7th and (m – 1)th numbers is 5:9. Find the value of m.
A man starts repaying a loan as first installment of Rs. 100. If he increases the installment by Rs 5 every month, what amount he will pay in the 30th installment?
If the sum of three numbers in A.P., is 24 and their product is 440, find the numbers.
Find the sum of all numbers between 200 and 400 which are divisible by 7.
A sequence is defined by an = n3 − 6n2 + 11n − 6, n ϵ N. Show that the first three terms of the sequence are zero and all other terms are positive.
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, ...
Which term of the A.P. 84, 80, 76,... is 0?
The 10th and 18th terms of an A.P. are 41 and 73 respectively. Find 26th term.
If (m + 1)th term of an A.P. is twice the (n + 1)th term, prove that (3m + 1)th term is twice the (m + n + 1)th term.
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.
If the sum of three numbers in A.P. is 24 and their product is 440, find the numbers.
Find the sum of the following serie:
2 + 5 + 8 + ... + 182
Find the sum of all odd numbers between 100 and 200.
Find the sum of all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7.
If Sn = n2 p and Sm = m2 p, m ≠ n, in an A.P., prove that Sp = p3.
If 12th term of an A.P. is −13 and the sum of the first four terms is 24, what is the sum of first 10 terms?
If a, b, c is in A.P., then show that:
b + c − a, c + a − b, a + b − c are in A.P.
If \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:
\[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
If a, b, c is in A.P., prove that:
a2 + c2 + 4ac = 2 (ab + bc + ca)
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.
In a cricket team tournament 16 teams participated. A sum of ₹8000 is to be awarded among themselves as prize money. If the last place team is awarded ₹275 in prize money and the award increases by the same amount for successive finishing places, then how much amount will the first place team receive?
Write the sum of first n even 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
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
If the first, second and last term of an A.P are a, b and 2a respectively, then its sum is
If in an A.P., Sn = n2p and Sm = m2p, where Sr denotes the sum of r terms of the A.P., then Sp is equal to
Write the quadratic equation the arithmetic and geometric means of whose roots are Aand G respectively.
If the sum of p terms of an A.P. is q and the sum of q terms is p, show that the sum of p + q terms is – (p + q). Also, find the sum of first p – q terms (p > q).
Let Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn then S3n: Sn is equal to ______.
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 ______.
If a1, a2, a3, .......... are an A.P. such that a1 + a5 + a10 + a15 + a20 + a24 = 225, then a1 + a2 + a3 + ...... + a23 + a24 is equal to ______.
The internal angles of a convex polygon are in A.P. The smallest angle is 120° and the common difference is 5°. The number to sides of the polygon is ______.
For an A.P. with first term \(a\) and common difference \(d\), the common difference is computed as:
