algebra

if 1, 2, 7 and 20, respectively, are added to the first terms of an arithmetic progression, a geometric progression of four terms is obtained. find the first term and common difference of the arithmetic progression

the answers are both 3 .. but i don't know the solution, please help me, thank you

asked by Sam
  1. let the first 4 terms of the AP be
    a-d, a , a+d, and a+2d

    new sequence is
    a-d+1, a+2, a+d + 7, and a+2d+20

    so (a+2)/(a-d+1) = (a+d+7)/(a+2)
    (a+2)^2 = (a-d+1)(a+d+7)
    a^2+4a+4 = a^2+ad+7a-ad-d^2-7d+7a+d+7
    -4a = -d^2 - 6d + 3
    4a = d^2 + 6d - 3 **

    (a+d+7)/(a+2) = (a+2d+20)/(a+d+7)
    (a+d+7)^2 = (a+2)(a+2d+20)
    a^2 + d^2 + 49 + 2ad +14a + 14d
    = a^2 + 2ad + 22a + 4d + 40

    8a = d^2 + 10d + 9 ***

    double **, then subtract ***
    0 = d^2 + 2d - 15 = 0
    (d + 5)(d-3) = 0
    d = -5, or d = 3

    if d = 3 in **
    4a = 3^2 + 18 - 3 = 24
    a = 6
    the original AP according to my definition , was 3, 6, 9, 12
    check: if I add the numbers as stated, I get 4, 8, 16, and 32 , which is a GP

    if d = -5, in **
    4a = 25 -30 - 3 = -8
    a = -2
    the original AP is 3, -2, -7, -12
    adding the numbers as stated will give me:
    4, 0, 0, 8 , but we can't have a 0 in a GP

    so we have to go with my first part of the solutions, which yielded the AP
    3, 6, 9, 12
    Making the first term 3, and the common difference as 3

    posted by Reiny

Respond to this Question

First Name

Your Response

Similar Questions

  1. Math

    1. What are the next two terms of the following sequence? -3, 1, 5, 9... 2. What are the next two terms of the following sequence? -2, 4, -8, 16... 3. What is the common difference of the following arithmetic sequence? 13, -7,
  2. Math Help!!!

    determine whether each sequence is arithmetic or geometric. find the next three terms. 1. 14,19,24,29.... geometric, 34,39,44 arithmetic,32,36,41 arithmetic 34,39,44**** the sequence is nether geometric nor arithmetic 2.
  3. math

    Determine whether each sequence is arithmetic or geometric. Find the next three terms. 1. 14, 19, 24, 29, . . . (1 point) geometric, 34, 39, 44 arithmetic, 32, 36, 41 arithmetic, 34, 39, 44 *** The sequence is neither geometric
  4. Algebra

    Determine whether each sequence is arithmetic or geometric. Find the next three terms. 14, 19, 24, 29, . . . A.geometric, 34, 39, 44 B.arithmetic, 32, 36, 41 C.arithmetic, 34, 39, 44 D.The sequence is neither geometric nor
  5. algebra 2

    An arithmetic progression has 5 terms that have a sum of 200. The sum of the last three terms added to 6 times the sum of the first two terms equals zero. What is the first term?
  6. 7th Grade Math for Steve.. or Ms. Sue

    Determine whether each sequence is arithmetic or geometric. Find the next three terms. 1. 14, 19, 24, 29, . . . (1 point) geometric, 34, 39, 44 arithmetic, 32, 36, 41 arithmetic, 34, 39, 44 ** The sequence is neither geometric nor
  7. arithmetic

    1. The first and last term of an A.P are, a and l respectively, show that the sum of nth term from the beginning and nth term from the end is a + l. 2. If mth term of an A.P be 1/n and nth term be 1/m, then show that its mnth term
  8. arithmetic

    1. The first and last term of an A.P are, a and l respectively, show that the sum of nth term from the beginning and nth term from the end is a + l. 2. If mth term of an A.P be 1/n and nth term be 1/m, then show that its mnth term
  9. arithmetic

    In an arithmetic series, the terms of the series are equally spread out. For example, in 1 + 5 + 9 + 13 + 17, consecutive terms are 4 apart. If the first term in an arithmetic series is 3, the last term is 136, and the sum is
  10. math

    The first, the third and the seventh terms of an increasing arithmetic progression are three consecutive terms of a geometric progression. In the first term of the arithmetic progression is 10 find the common difference of the

More Similar Questions