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Find the 6th and nth terms of the GP 2, 6, 18, 54….
Find the 17th and nth terms of the GP 2, 2√2, 4, 8√2….
Find the 7th and nth terms of the GP 0.4, 0.8, 1.6….
Find the 10th and nth terms of the GP
Which term of the GP 3, 6, 12, 24…. Is 3072?
Which term of the GP is -128?
Which term of the GP√3, 3, 3√3… is 729?
Find the geometric series whose 5th and 8th terms are 80 and 640 respectively.
Find the GP whose 4th and 7th terms are and respectively.
The 5th, 8th and 11th terms of a GP are a, b, c respectively. Show that b2 = ac
The first term of a GP is -3 and the square of the second term is equal to its 4th term. Find its 7th term.
Find the 6th term from the end of GP 8, 4, 2….
Find the 4th term from the end of the GP.
If a, b, c are the pth, qth and rth terms of a GP, show that
(q – r) log a + (r – p) log b + (p – q) log c = 0.
The third term of a GP is 4; Find the product of its five terms.
In a finite GP, prove that the product of the terms equidistant from the beginning and end is the product of first and last terms.
If then show that a, b, c, d are in GP.
If a and b are the roots of x2 – 3x + p = 0 and c and d are the roots of x2 – 12x + q = 0, where a, b, c, d from a GP, prove that (q + p): (q – p) = 17: 15.
Find the sum of the GP :
1 + 3 + 9 + 27 + …. To 7 terms
1 + + 3 + 3+….. to 10 terms
0.15 + 0.015 + 0.0015 + …. To 6 terms
to 9 terms
……to 8 terms
…. To 6 terms
+ … to n terms
… to n terms
1 – a + a2 – a3 + …to n terms ( a ≠ 1)
x3 + x5 + x7 + …. To n terms
x(x + y) + x2(x2 + y2) + x3(x3 + y3) + …. To n terms
Find the sum to n terms of the sequence :
(i) ,….. to n terms
(ii) (x + y), 9x2 + xy + y2), (x3 + x2y + xy2 + y3), …. to n terms
Find the sum :
…. To 2n terms
NOTE:In an expression like this ⇒ , n represents the upper limit, 1 represents the lower limit , x is the variable expression which we are finding out the sum of and i represents the index of summarization.
Find the sum of the series :
NOTE: The following terms are not G.P. series, but we can convert them to form one.
(i) 8 + 88 + 888 + …. To n terms
(ii) 3 + 33 + 333 + …. To n terms
(iii) 0.7 + 0.77 + 0.777 + …. To n terms
The sum of n terms of a progression is (2n – 1). Show that it is a GP and find its common ratio.
In a GP, the ratio of the sum of the first three terms is to first six terms is 125 : 152. Find the common ratio.
Find the sum of the geometric series 3 + 6 + 12 + … + 1536.
How many terms of the series 2 + 6 + 18 + …. + must be taken to make the sum equal to 728?
The common ratio of a finite GP is 3, and its last term is 486. If the sum of these terms is 728, find the first term.
The first term of a GP is 27, and its 8th term is . Find the sum of its first 10 terms.
The 2nd and 5th terms of a GP are and respectively. Find the sum of n terms GP up to 8 terms.
The 4th and 7th terms of a GP are and respectively. Find the sum of n terms of the GP.
A GP consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying the odd places, find the common ratio of the GP.
Show that the ratio of the sum of first n terms of a GP to the sum of the terms from (n + 1)th to (2n)th term is .
What will 15625 amount to in 3 years after its deposit in a bank which pays annual interest at the rate of 8% per annum, compounded annually?
The value of a machine costing 80000 depreciates at the rate of 15% per annum. What will be the worth of this machine after 3 days?
Three years before the population of a village was 10000. If at the end of each year, 20% of the people migrated to a nearby town, what is its present population?
What will 5000 amount to in 10 years, compounded annually at 10% per annum? [Given (1.1)10 = 2.594]
A manufacturer reckons that the value of a machine which costs him 156250, will depreciate each year by 20%. Find the estimated value at the end of 5 years.
The number of bacteria in a certain culture doubles every hour. If there were 50 bacteria present in the culture originally, how many bacteria would be present at the end of (i) 2nd hour, (ii) 5th hour and (iii) nth hour?
If p, q, r are in AP, then prove that pth, qth and rth terms of any GP are in GP.
If a, b, c are in GP, then show that log an, log bn, log cn are in AP.
If a, b, c are GP, then show that , are in AP.
Find the values of k for which k + 12, k – 6 and 3 are in GP.
Three numbers are in AP, and their sum is 15. If 1, 4, 19 be added to them respectively, then they are in GP. Find the numbers.
Three numbers are in AP, and their sum is 21. If the second number is reduced by 1 and the third is increased by 1, we obtain three numbers in GP. Find the numbers.
The sum of three numbers in GP is 56. If 1, 7, 21 be subtracted from them respectively, we obtain the numbers in AP. Find the numbers
If a, b, c are in GP, prove that .
If (a – b), (b – c), (c – a) are in GP then prove that (a + b + c)2 = 3(ab + bc + ca).
If a, b, c are in GP, prove that
(i) a(b2 + c2) = c(a2 + b2)
(iii) (a + 2b + 2c)(a – 2b + 2c) = a2 + 4c2
If a, b, c, d are in GP, prove that
(i) (b + c)(b + d) = (c + a)(c + a)
(iii) (a + b + c + d)2 = (a + b)2 + 2(b + c)2 + (c + d)2
If a, b, c are in GP, prove that are in AP.
If a, b, c are in GP, prove that a2, b2, c2 are in GP.
If a, b, c are in GP, prove that a3, b3, c3 are in GP
If a, b, c are in GP, prove that (a2 + b2), (ab + bc), (b2 + c2) are in GP.
If a, b, c, d are in GP, prove that (a2 – b2), (b2 – c2), (c2 – d2) are in GP.
If a, b, c, d are in GP, then prove that
are in GP
If (p2 + q2), (pq + qr), (q2 + r2) are in GP then prove that p, q, r are in GP
If a, b, c are in AP, and a, b, d are in GP, show that a, (a – b) and (d – c) are in GP.
If a, b, c are in AP, and a, x, b and b, y, c are in GP then show that x2, b2, y2 are in AP.
Find two positive numbers a and b, whose
(i) AM = 25 and GM = 20
(ii) AM = 10 and GM = 8
Find the GM between the numbers
(i) 5 and 125
(ii) 1 and
(iii) 0.15 and 0.0015
(iv) -8 and -2
(v) -6.3 and -2.8
(vi) ad ab3
Insert two geometric means between 9 and 243.
Insert three geometric means between and 432.
Insert four geometric means between 6 and 192.
The AM between two positive numbers a and b(a>b) is twice their GM. Prove that a:b .
If a, b, c are in AP, x is the GM between a and b; y is the GM between b and c; then show that b2 is the AM between x2 and y2.
Show that the product of n geometric means between a and b is equal to the nth power of the single GM between a and b.
If AM and GM of the roots of a quadratic equation are 10 and 8 respectively then obtain the quadratic equation.
Find the sum of each of the following infinite series :
6 + 1.2 + 0.24 + ….. ∞
10 – 9 + 8.1 - ……∞
Prove that 91/3 × 91/9 × 91/27 × …..∞ = 3
Find the rational number whose decimal expansion is given below :
Express the recurring decimal 0.125125125 …. = as a rational number.
Write the value of in the form of a simple fraction.
The sum of an infinite geometric series is 6. If its first term is 2, find its common ratio.
The sum of an infinite geometric series is 20, and the sum of the squares of these terms is 100. Find the series.
The sum of an infinite GP is 57, and the sum of their cubes is 9747. Find the GP.
If the 5th term of a GP is 2, find the product of its first nine terms.
If the (p + q)th and (p – q)th terms of a GP are m and n respectively, find its pth term.
If 2nd, 3rd and 6th terms of an AP are the three consecutive terms of a GP then find the common ratio of the GP.
Write the quadratic equation, the arithmetic and geometric means of whose roots are A and G respectively.
If a, b, c are in GP and a1/x = b1/y = c1/z then prove that x, y, z are in AP.
If a, b, c are in AP and x, y, z are in GP then prove that the value of xb - c. yc - a. za - b is 1.
Express as a rational number.
Express as a rational number.
The second term of a GP is 24 and its fifth term is 81. Find the sum of its first five terms.
The ratio of the sum of first three terms is to that of first six terms of a GP is 125 : 152. Find the common ratio.
The sum of first three terms of a GP is and their product is 1. Find the common ratio and these three terms.