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Zorluk: ZorArithmetic and Geometric Sequences and Series

The first, third, and eleventh terms of a non-constant arithmetic sequence are the first, second, and third terms, respectively, of a geometric sequence. If the first term of the arithmetic sequence is 66, what is the sum of the first 44 terms of the geometric sequence?

  1. A
    78
  2. B
    384
  3. 510Cevap
  4. D
    1,554
  5. E
    4,920

Cevap

The sum of the first 4 terms of the geometric sequence is 510.
By writing the first, third, and eleventh terms of the arithmetic sequence as 66, 6+2d6+2d, and 6+10d6+10d, we can set up the geometric sequence relation (6+2d)2=6(6+10d)(6+2d)^2 = 6(6+10d). Solving this quadratic equation for dd yields d=9d=9 (since the sequence is non-constant, d≠0d \neq 0). Substituting d=9d=9 back gives the first three geometric terms as 66, 2424, and 9696, which means the common ratio rr is 44. The sum of the first 44 terms of this geometric sequence is calculated as 6(44−1)/(4−1)=5106(4^4-1)/(4-1) = 510.

Adım Adım Çözüm

1
Define the terms of the arithmetic and geometric sequences.
Let the arithmetic sequence have first term a1=6a_1 = 6 and common difference dd. The first, third, and eleventh terms are a1=6a_1 = 6, a3=6+2da_3 = 6 + 2d, and a11=6+10da_{11} = 6 + 10d. These are the first three terms of the geometric sequence: g1=6g_1 = 6, g2=6+2dg_2 = 6 + 2d, and g3=6+10dg_3 = 6 + 10d.
This establishes algebraic expressions for the terms based on their positions in the sequences.
2
Set up a relation using the constant ratio of the geometric sequence and solve for dd.
Since g1g_1, g2g_2, and g3g_3 form a geometric sequence, (g2)2=g1⋅g3(g_2)^2 = g_1 \cdot g_3. Substituting the expressions gives (6+2d)2=6(6+10d)  ⟹  36+24d+4d2=36+60d  ⟹  4d2−36d=0(6 + 2d)^2 = 6(6 + 10d) \implies 36 + 24d + 4d^2 = 36 + 60d \implies 4d^2 - 36d = 0. Since the sequence is non-constant (d≠0d \neq 0), we divide by 4d4d to get d=9d = 9.
Solving the equation yields the common difference of the arithmetic sequence.
3
Determine the terms and common ratio of the geometric sequence.
Using d=9d = 9, the first two terms of the geometric sequence are g1=6g_1 = 6 and g2=6+2(9)=24g_2 = 6 + 2(9) = 24. The common ratio is r=246=4r = \frac{24}{6} = 4.
Finding the common ratio allows the use of the geometric series sum formula.
4
Compute the sum of the first 4 terms of the geometric sequence.
S4=g1r4−1r−1=644−14−1=6256−13=2(255)=510S_4 = g_1 \frac{r^4 - 1}{r - 1} = 6 \frac{4^4 - 1}{4 - 1} = 6 \frac{256 - 1}{3} = 2(255) = 510.
This calculates the final required sum.

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Arithmetic and Geometric Sequences and Series
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