1) n arithmetic means are inserted between 1 and 41. If the sum of the means and the two numbers be 189, find the value of n.
2) If (b + c)/a, (c + a)/b, (a+ b)/c are in AP and a+ b + c ≠ 0, show that 1/am 1/b, 1/c are in AP.
3) If b²+ bc + c², c²+ ca + a², a²+ ab + b² are in AP, show that a, b, c are in AP.
4) If a, b, c are in AP show that (1/a) (1/b + 1/c), (1/b)(1/c + 1/a), (1/c) (1/a + 1/b) are in AP.
5) If a,b,c are in AP, show that ab + cd + ad = 3bc.
6) A farmer undertakes to pay off a debt of Rs 2700 by monthly installments. He pays Rs 200 as the first instalment and increases every subsequent installment by Rs 25 over the immediate previous instalment. In how many installment his debt will be cleared up ?
7) The sum of first n terms of an AP is denoted by Sₙ. If Sₙ = n²p and Sₘ = m²p, m≠ n, then show that, Sₚ = p³.
8) There are 100 terms in an AP. If the sum of the terms occupying the even places is 1600 and the sum of the terms occupying the odd places is 1300. Find the two middle terms of the AP.
9) If the sum of first p terms of an AP be P and the sum of first q terms be Q, show that the first term of the progression will be {Pq(q -1) - Qp(p -1)}/{pq(q - p)}.
10) If the first and the last terms of an AP are a and l respectively and the sum of the terms is S, show that the common difference is (l² - a²)(2S - (l + a)).
11) The sum of three numbers in GP is 7 and the sum of their squares is 21, find the sum of their cubes.
12) The sum of three positive numbers in GP is 65 and the product of the first and the third number is 225. Find the numbers.
13) Find, without assuming the summation formula, the sum of the GP
2+4+8+.....to 20 terms.
14) If (a + bx)/(a - bx)= (b + cx)/(b - cx)= (c + dx)/(c - dx) (x ≠ 0), prove that a, b, c, d are in GP.
15) If a,b,c,d are in GP, show that, 1/(a² + b²) , 1/(b² + c²), 1/(c² + d²) are in GP.
16) The 5th term of a GP is 8 and the 13th term is 128. Find the sum upto 12th term.
17) The sum of three consecutive terms of an AP is 15; if 1 is subtracted from the second term, the resulting numbers form a GP. Find the terms.
18) The second, third and sixth terms of an AP are three consecutive terms of a GP. Find the common ratio of the GP.
19) Find the sum of:
a) 1+11+111+1111+ .....to n terms.
b) 0.8+0.88+0.888+ .....to n terms.
c) 1+ 4+ 10+22+ ....to n terms.
d) 1+ (1+2) + (1+2+2²)+.....to n terms.
e) 1+ (1+ x)+ (1+ x + x²)+.....to n terms.
f) 1/2+ 3/2²+ 5/2³+ ......+ (2n -1)/2ⁿ.
20) If the first, second and last terms of a GP are a, b and l respectively show that the sum of the terms is (bl - a²)/(l - a).
21) If u₁, u₂, u₃, .... are in GP whose common ratio is k, then express the value of u₁u₂ + u₂u₃ + .....+ uₙuₙ₊₁ in terms of k and u₁.
22) If pth, qth , rth terms of an AP are in GP, then show that the common ratio of the GP is (q - r)/(p - q).
23) If the pth, qth, rth and sth terms of an AP are in GP, then show that p - q, q - r, r - s are in GP.
24) Solve: 1+ 3+ 3²+......3ˣ = 1093.
25) If a, b, c, d are in AP and a, c, d are in GP, show that a² - d² = 3(b² - ad).
26) The product of three numbers in GP is 512. If 8 be added to the first and 6 to the second, the resulting numbers with the third number from form an AP. Find the numbers.
27) If a, b, c are in AP and a, b- a, c - a are in GP, show that a= b/3= c/5.
28) If (a - x)/px = (a - y)/qy = (a - z)/rz and p, q, r are three consecutive terms of an AP, show that 2/y = 1/x + 1/z.
29) There are even number of terms in a GP. If the sum of the progression is thrice the sum of the odd terms, then find the common ratio of the GP.
30) Prove that the product of n geometric means between a and b is (ab)ⁿ⁾².
31) The arithmetic mean of two numbers is A and their geometric mean is G; find the numbers.
32) Find the least value of n for which 1+ 3+ 3² + ....+ 3ⁿ⁻¹ > 900.
33) If tₙ be the nth term of a GP and t₁ = 2, find the minimum value of 2t₂ + 3t₃. Find also the value of the common ratio for which the expression will be minimum.
34) A man saves Rs 4 in the first month. Now from the second month he saves in every month twice the savings of the preceding month. Thus after 8 months, from the 9th month he saves in every month Rs 4 less than the savings of the immediate previous month. What will be his total savings in 16 months.
35) If logᵧx, logᵥy, logₓv are in GP and also in AP, show that, x= y = z.
36) Simplify:
a) 20/(√3 - √-2) + 30/(3√-2 - 2√3) - 14/(2√3 - √-2).
b) (cos30°+ i sin30)⁴/(cos60 + i sin 60)². 1
37) If {(1+ i)/(1- i)}³ - {(1- i)/(1+ i)}³ = p + iq, find (p, q).
38) If x= 2+ 3i and y= 2- 3i, find the value of (x³ - y³)/(x³ + y³).
38) If x= 1+ 3i, find the value of x⁴- 5x³ + 18x² - 34x +2.
39) If x= - 5+ 2√-4, find the value of x⁴+ 9x³+ 35x² - x +4.
40) If x= 2+ i, find the value of x³ - 5x²+ 9x -5.
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