Question 1
a) Find 3 arithmetic means between 2 and 10.
Answer: 4, 6, 8
b) Find 4 arithmetic means between 2 and 27.
Answer: 7, 12, 17, 22
Question 2
Find the sum of 500 AM's between 2 and 3.
Answer: 1250
Solution Hint: 2, A1, A2, ……., A500 , 3 are in AP
a = 2, A502 = 3 ⇒ d = 1/501
A1, A2, ……., A500 = 500/2 ( A1 + A500 )
= 250 (2 + d + 3 - d) = 1250
Question 3
Between
5 and 35, m numbers have been inserted in such a way that the resulting
sequence is an A.P. and the ratio of 3rd and (m - 2)th numbers is 7 : 13. Find
the value of m.
Ans:
m = 9
Question 4
Which term of the GP : 2, 2√2, 4, ..... is 8 ?
Answer: 5th term.
Question 5
The product of the first three terms of a G.P. is 1000. If we add 6 to its second term and 7 to its third term, the resulting three terms form an A.P. Find the terms of the G.P.
Ans: 5, 10, 20 or 20, 10, 5
Question 6
Find three numbers in GP whose sum is 7 and product is 8.
Answer: 4, 2, 1 or 1, 2, 4
Question 7
If a and b are the roots of x2 - 3x + p = 0 and c, d are the roots of x2 - 12x + q = 0, where a, b, c, d forma G.P. Prove that (q + p) : (q - p) = 17: 15.
Solution Hint: r2 = 4
Question 8
If the first term of GP is a and nth term is b and P denotes the product of first n terms. Prove that P2 = (ab)2
Question 9
If p, q, r are in GP and the equations px2 +2qx + r = 0 and dx2 +2ex + f = 0 have a common root then show that d/p, e/q, f/r are in A.P.
Question 10
Find the sum of n terms of the series:
0.6 + 0.66 + 0.666 + ..............
Ans:
Question 11
The sum of two positive numbers is 4 times their geometric mean, show that numbers are in the ratio (2 + √3 ) : (2 – √3 ).
OR
Find the ratio of two positive numbers a and b such that AM: GM = 2 : 1
Ans: (2 + √3 ) : (2 – √3 ).
OR
The AM of two positive numbers is 3 times their GM. Prove that numbers are in the ratio (3+2√2) : (3-2√2)
Question12
Find the sum to infinity of the GP: 4, 4/3, 4/9, ......
Answer: 6
Question 13
Aman saw his younger brother playing with building blocks. he observed that he is playing in a pattern by making a tower of 32 blocks and then dividing into half a tower of 16 blocks and so on still he got a tower of 1 block.
a) What kind of pattern is brother making? Also, how many Towers did the boy make?
Answer: Brother making the towers of blocks: 32, 16, 8, 4, 2, 1 and it is in GP
b) How many total blocks does the boy carry with him?
Answer: 63
Solution Hint
Total towers made by the boy = 6
Here a = 32, r = 1/2, n = 6, So find S6
Question 14
On a Sunday Sunil and his friend went to see a circus show and observed
that the arrangement of chairs in a row has a Peculiar arrangement. The number
of chairs in his row, which is the third row, are 125 and the number of chairs
in the First row are 75. The number of chairs is increasing by a fixed number.
There are 12 rows of chairs in all.
a) How many chairs are there in the 7th row, where his friend is situated?
Ans: a = 75, a3 = 125
d = 25
a7 = a + 6d = 225
b) If all the chairs are occupied, how many people saw that show? also
find the Arithmetic Mean of the number of chairs in the row Sunil and his
friend was seated?
Ans: S12 = 2550
AM of 125 and 225 = 175
In a cricket tournament 16 school teams participated. A sum of ` 8000 is to be awarded among themselves as prize money. If the last placed team is awarded ` 275 in prize money and the award increases by the same amount for successive finishing places, how much amount will the first place team receive?
Ans: ₹ 725
Question 17
If the third term of GP is 4, then the product of its first 5 terms is
Ans: (4)5.
Solution Hint: It is given that, T3 = 4
Let a and r the first term and common ratio, respectively.
Then, ar2 = 4 ...(i)
Product of first 5 terms = ⋅ ⋅ ⋅ ⋅ a (ar) (ar2 ) (ar3 ) (ar4 )
= a5 r10 = (ar2)5 = (4)5. ....... using (i)
Question 18
Find two numbers whose A.M. is 25 and G.M. is 20
Answer: 40 and 10
Question 19
If arithmetic mean and geometric mean of two positive numbers a and b are 10 and 8 respectively, find the numbers.
Answer: 4, 16 or 16, 4
Question 20
Find the value of (32)(32)1/6(32)1/36 ............. ∞
Question 21
In a set of four numbers, the first three are in GP and the last three are in AP with a common difference of 6. If the first and fourth terms are equal, find the four numbers.
Ans: 8, - 4, 2, 8
Explanation: As the last three numbers are in A.P. with a common difference of 6
Let the last three numbers be b, b + 6 and b + 12.
Let a be the first number, then the four numbers in order are a, b, b + 6 and b + 12.
According to the question, the first three numbers are in G.P.
b2 = a (b + 6) …….. (1)
Also first and fourth numbers are equal, so
a = b + 12 …………. (ii)
Substituting the value of a from (ii) in (i), we get
b2 = (b + 12) (b + 6)
b2 = b2 + 18b + 72
18b + 72 = 0
b = - 4
From (ii), we get a = - 4 + 12 = 8
Hence, the required numbers are 8, - 4, 2, 8
Question 22
If the sum of an infinite geometric series is 15 and the sum
of the squares of the terms of the series is 45, find the G.P.
Ans:
Solution Hint:
Using the formula for sum to infinite terms
ATQ
sum of an infinite geometric series is 15
⇒ a = 15 - 15r .......(i)
sum of the squares of the infinite terms of the series is 45,
Question 1
If 9 times the 9th term of an AP is equal to 13 times the 13th term, then find 22nd term of the AP?
Ans: Zero
Question 2
If sum of n terms of two AP’s are in the ratio 3n + 8 : 7n + 15, then find the ratio of their 12th terms
Ans: 77/176
Question 3
Find the sum of n terms of the series
3 + 5 + 9 + 15 + 23 + …………………. N terms
Ans: n2+2n
Question 4
If a, b and c are in A.P and ab + bc + ca = 0 then prove that a²(b + c), b²(a + c), c2(a + b) are also in A.P.
Question 5
Find the sum of the following series up to n terms :
Ans:
Question 6
The ratio of sums of m and n terms of an A.P. is m2 : n2. Show that ratio of mth and nth term is (2m – 1) : (2n – 1)
Ans: 6r - 1
Question 9
If Sn denotes the sum of first n terms of an AP, If S2n = 3Sn, then find S3n : Sn
Ans: 6 : 1
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