Questions on Algebra: Sequences of numbers, series and how to sum them answered by real tutors!

Tutors Answer Your Questions about Sequences-and-series (FREE)

Question 1185679: A frog jumped 20 centimeters away from a door and then jumps again 15
centimeters away from the 20-centimeter point. The frog reaches three-
fourths of its preceding distance each time it jumps.
Show relationship between the number of jumps and distance covered by the frog per jump. Then, write the geometric sequence formed and answer the question that follows.
Number of jumps and distance covered:
1 -
2 -
3 -
4 -
5 -
6 -
7 -
Geometric Sequence:
Question: If the frog jumped following a straight path, what is the linear distance of the frog from the door after jumping seven times?

Click here to see answer by greenestamps(13200)

Question 1184306: Let a, b, c, p, q, r be positive real numbers such that a, b, c are in geometric sequence and then which one of the following condition holds:
A- p, q, r are in geometric sequence
B- p, q, r are in arithmetic sequence
C- p, q, r are in harmonic sequence
D- p^2, q^2, r^2 are in arithmetic sequence
E- p^2, q^2, r^2 are geometric sequence
..
[Note: ^2 means power 2]
Click here to see answer by CPhill(1959)

Question 1209756: . A construction company purchases a bulldozer for $160.000. Each year the value of the bulldozer depreciates by 20% of its value in the preceding year. Let V, be the value of the bulldozer in the nth year. (Let n = 1 be the year the bulldozer is purchased)
a) Find the formula for V.
12 Marks)
b) In what year will the value of the bulldozer be less than $100.000
(4 Marks)
Click here to see answer by CPhill(1959)

Question 1179819: f. Find the present values of the following annuities
i. RM6,000 every year for 8 years at 12% compounded annually
ii. RM800 every month for 2 years 5 months at 5% compounded monthly

Click here to see answer by CPhill(1959)

Question 1179223: 1)Janie and Jasmine are playing three games at an arcade. Each of the games requires either 2, 3, or 4 tokens. The girls plan to play as many games as they can before running out of tokens. Write an expression to represent the total number of tokens that Janie and Jasmine will need to play each of the three games at least once. Let m represent the number of games that require 2 n tokens; represent the number of games that require 3 p tokens, and represent the number of games that require 4 tokens.

2)Janie and Jasmine are playing three games at an arcade. Each of the games requires either 2, 3, or 4 tokens. The girls plan to play as many games as they can before running out of tokens. Let m represent the number of games that require 2 tokens; n represents the number of games that require 3 tokens, and p represents the number of games that require 4 tokens. Janie plays the 3-token game four times and Jasmine plays the 4-token games 5 times. Write two equivalent expressions to represent the number of tokens that the girls will need to play each of the three games at least one time.

Click here to see answer by CPhill(1959)

Question 1179200: Give the first five terms of the sequences?
1. a 1 =-4;a n =2a n-1 -1 for (n>=2)
2. a 1 = 2/5 ,a 2 = 4/5 ;a n =a n-2 * a n-1 ( for n>=3)
Click here to see answer by CPhill(1959)

Question 1209782: Evaluate
2 + \frac{6}{7} +\frac {18}{49} + \frac{54}{343} + 1 + \frac{2}{14} + \frac{3}{98}.
Click here to see answer by josgarithmetic(39617)

Question 1209780: Evaluate 21 + 27 + 33 + ... + 261 + 267 + 273 + ... + 6021.
Click here to see answer by greenestamps(13200)

Question 1209780: Evaluate 21 + 27 + 33 + ... + 261 + 267 + 273 + ... + 6021.
Click here to see answer by math_tutor2020(3817)

Question 1209783: Evaluate
1 + \frac{i}{3} - \frac{1}{9} - \frac{i}{27} + \frac{1}{81},
where $i$ is the imaginary unit.

Click here to see answer by greenestamps(13200)

Question 1209784: Evaluate \frac{3}{2} + \frac{5}{4} + \frac{9}{8} + \frac{17}{16}.
Click here to see answer by josgarithmetic(39617)

Question 1209784: Evaluate \frac{3}{2} + \frac{5}{4} + \frac{9}{8} + \frac{17}{16}.
Click here to see answer by math_tutor2020(3817)

Question 1209781: Evaluate \sum_{n = 1}^{33} (5n + 2 - 4n + n^2 + 17).
Click here to see answer by math_tutor2020(3817)

Question 1209793: The sum of the first three terms of a geometric sequence of integers is equal to seven times the first term, and the sum of the first four terms is $30$. What is the first term of the sequence?
Click here to see answer by greenestamps(13200)

Question 1209795: Find the sum of the series
$$1 + \frac{1}{2} + \frac{1}{10} + \frac{1}{20} + \frac{1}{100} + \cdots,$$
where we alternately multiply by $\frac 12$ and $\frac 15$ to get successive terms.
Click here to see answer by CPhill(1959)

Question 1177273: a(n) stands for sequence
a(1)=2, a(2)=3,
Prove that the limit
when n->∞,
exists and evaluate it
Click here to see answer by CPhill(1959)

Question 1177009: Prove that 5^{3^n} + 1 is divisible by 3^{n + 1} for all nonnegative integers n. Prove this using induction. Can you please explain in detail? Thank you so much!
Click here to see answer by CPhill(1959)

Question 1176853: Does the series |-1 + 1/e^n| converges or conditionally converges where n is from 0 to infinity?
Click here to see answer by CPhill(1959)

Question 1176741: The sequence (a_n) is defined by a_1 = {1}/{2} and
a_n = a_{n - 1}^2 + a_{n - 1}for n <= 2.
Prove that
{1}/{a_1 + 1} + {1}/{a_2 + 1} + ... + {1}/{a_n + 1} < 2 for all n <= 1.
Click here to see answer by CPhill(1959)

Question 1209805: Let a_1 + a_2 + a_3 + dotsb be an infinite geometric series with positive terms. If a_2 = 10, then find the smallest possible value of
a_1 + a_2 + a_3.
Click here to see answer by CPhill(1959)

Question 1209803: Let
a + ar + ar^2 + ar^3 + \dotsb
be an infinite geometric series. The sum of the series is 9. The sum of the cubes of all the terms is 36. Find the common ratio.
Click here to see answer by CPhill(1959)

Question 1209814: Let r be a real number such that |r| < 1. Express
\sum_{n = 0}^{\infty} n*r^n*(n + 1)*(n + 2)
in terms of r.
Click here to see answer by CPhill(1959)

Question 1209813: Find
\sum_{k = 0}^{10} (k + 3) \cdot 2^k \cdot (k - 3)
Click here to see answer by CPhill(1959)

Question 1209812: Simplify \frac{1 + 3 + 5 + ... + 1999 + 2001 + 2003}{2 + 4 + 6 + ... + 2000 + 2002 + 2004 + 2006 + 2008 + 2010}.
Click here to see answer by CPhill(1959)

Question 1209818: .
Click here to see answer by CPhill(1959)

Question 1209818: .
Click here to see answer by ikleyn(52787)

Question 1209818: .
Click here to see answer by greenestamps(13200)

Question 1209819:
Click here to see answer by ikleyn(52787)

Question 1209821:
Click here to see answer by ikleyn(52787)

Question 1209820:
Click here to see answer by ikleyn(52787)

Question 1209826: For a positive integer k, let
S_k = 1 \cdot 1! \cdot 2 + 2 \cdot 2! \cdot 3 + \dots + k \cdot k! \cdot (k + 1).
Find a closed form for S_k.
Click here to see answer by CPhill(1959)

Question 1209824: An estate developer had a forty year building plan for a country starting from January 2001. The number of houses to be built each year forms an arithmetic progression (A.P). 372 houses were built in 2010 and plans to build 1032 houses in 2040. Find the:
(a) number of houses expected to be built in 2023;
(b) total number of houses expected from the developer at the end of forty years.
Click here to see answer by CPhill(1959)

Question 1209822: Find
\sum_{k = 1}^{20} k(k^2 - 10k - 20)(k^2 + 1)
Click here to see answer by CPhill(1959)

Question 1209827: Find a closed form for
S_n = 1! \cdot (1^2 + 1) + 2! \cdot (2^2 + 2) + \dots + n! \cdot (n^2 + n).\]
for any integer n \ge 1. Your response should have a factorial.
Click here to see answer by CPhill(1959)

Question 1209829: Find the ordered pair (p,q) such that
F_n = p \alpha^n + q \beta^n.
Click here to see answer by CPhill(1959)

Question 1209830: Let
A_0 = 0
A_1 = 1
A_n = A_{n - 1} + A_{n - 2} for n ge 2
There is a unique ordered pair (c,d) such that c \alpha^n + d \beta^n is the closed form for sequence A_n. Find the ordered pair (c,d).

Click here to see answer by CPhill(1959)

Question 1209828: The Fibonacci sequence, is defined by F_0 = 0, F_1 = 1, and F_n = F_{n - 2} + F_{n - 1}. It turns out that
F_n = \frac{\alpha^n - \beta^n}{\sqrt{5}},
where \alpha = \frac{1 + \sqrt{5}}{2} and \beta = \frac{1 - \sqrt{5}}{2}.

The Lucas sequence is defined as follows: L_0 = 2, L_1 = 1, and
L_n = L_{n - 1} + L_{n - 2}
for n \ge 2. What is L_4?

Click here to see answer by CPhill(1959)

Question 1209833: Fill in the blanks, to make a true equation.

3/(3^2 - 1) + 3^2/(3^4 - 1) + 3^3/(3^6 - 1) + 3^4/(3^8 - 1) + ... + 3^(2(n - 1))/(3^(2n) - 1) = ___/___
Click here to see answer by ikleyn(52787)

Question 1209845: The second and seventh term of a geometric progression are 18 and 4374 respectively. Find the sum of the fourth and the eighth term when the difference is 3

Click here to see answer by CPhill(1959)

Older solutions: 1..45, 46..90, 91..135, 136..180, 181..225, 226..270, 271..315, 316..360, 361..405, 406..450, 451..495, 496..540, 541..585, 586..630, 631..675, 676..720, 721..765, 766..810, 811..855, 856..900, 901..945, 946..990, 991..1035, 1036..1080, 1081..1125, 1126..1170, 1171..1215, 1216..1260, 1261..1305, 1306..1350, 1351..1395, 1396..1440, 1441..1485, 1486..1530, 1531..1575, 1576..1620, 1621..1665, 1666..1710, 1711..1755, 1756..1800, 1801..1845, 1846..1890, 1891..1935, 1936..1980, 1981..2025, 2026..2070, 2071..2115, 2116..2160, 2161..2205, 2206..2250, 2251..2295, 2296..2340, 2341..2385, 2386..2430, 2431..2475, 2476..2520, 2521..2565, 2566..2610, 2611..2655, 2656..2700, 2701..2745, 2746..2790, 2791..2835, 2836..2880, 2881..2925, 2926..2970, 2971..3015, 3016..3060, 3061..3105, 3106..3150, 3151..3195, 3196..3240, 3241..3285, 3286..3330, 3331..3375, 3376..3420, 3421..3465, 3466..3510, 3511..3555, 3556..3600, 3601..3645, 3646..3690, 3691..3735, 3736..3780, 3781..3825, 3826..3870, 3871..3915, 3916..3960, 3961..4005, 4006..4050, 4051..4095, 4096..4140, 4141..4185, 4186..4230, 4231..4275, 4276..4320, 4321..4365, 4366..4410, 4411..4455, 4456..4500, 4501..4545, 4546..4590, 4591..4635, 4636..4680, 4681..4725, 4726..4770, 4771..4815, 4816..4860, 4861..4905, 4906..4950, 4951..4995, 4996..5040, 5041..5085, 5086..5130, 5131..5175, 5176..5220, 5221..5265, 5266..5310, 5311..5355, 5356..5400, 5401..5445, 5446..5490, 5491..5535, 5536..5580, 5581..5625, 5626..5670, 5671..5715, 5716..5760, 5761..5805, 5806..5850, 5851..5895, 5896..5940, 5941..5985, 5986..6030, 6031..6075, 6076..6120, 6121..6165, 6166..6210, 6211..6255, 6256..6300, 6301..6345, 6346..6390, 6391..6435, 6436..6480, 6481..6525, 6526..6570, 6571..6615, 6616..6660, 6661..6705, 6706..6750, 6751..6795, 6796..6840, 6841..6885, 6886..6930, 6931..6975, 6976..7020, 7021..7065, 7066..7110, 7111..7155, 7156..7200, 7201..7245, 7246..7290, 7291..7335, 7336..7380, 7381..7425, 7426..7470, 7471..7515, 7516..7560, 7561..7605, 7606..7650, 7651..7695, 7696..7740, 7741..7785, 7786..7830, 7831..7875, 7876..7920, 7921..7965, 7966..8010, 8011..8055, 8056..8100, 8101..8145, 8146..8190, 8191..8235, 8236..8280, 8281..8325, 8326..8370, 8371..8415, 8416..8460, 8461..8505, 8506..8550, 8551..8595, 8596..8640, 8641..8685, 8686..8730, 8731..8775, 8776..8820, 8821..8865, 8866..8910, 8911..8955, 8956..9000, 9001..9045, 9046..9090, 9091..9135, 9136..9180, 9181..9225, 9226..9270, 9271..9315, 9316..9360, 9361..9405, 9406..9450, 9451..9495, 9496..9540, 9541..9585, 9586..9630, 9631..9675, 9676..9720, 9721..9765, 9766..9810, 9811..9855, 9856..9900, 9901..9945, 9946..9990, 9991..10035, 10036..10080, 10081..10125, 10126..10170, 10171..10215, 10216..10260, 10261..10305, 10306..10350, 10351..10395, 10396..10440, 10441..10485, 10486..10530, 10531..10575, 10576..10620, 10621..10665, 10666..10710, 10711..10755, 10756..10800, 10801..10845, 10846..10890, 10891..10935, 10936..10980, 10981..11025, 11026..11070, 11071..11115, 11116..11160, 11161..11205, 11206..11250, 11251..11295, 11296..11340, 11341..11385, 11386..11430, 11431..11475, 11476..11520, 11521..11565, 11566..11610, 11611..11655, 11656..11700, 11701..11745, 11746..11790