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Zorluk: OrtaArithmetic and Geometric Progressions (AP and GP)

Arrange the following values related to arithmetic and geometric progressions in ascending order (from smallest to largest):

  1. 1The common ratio rr of a geometric progression with positive terms whose 2nd2^{\text{nd}} term is 55 and 4th4^{\text{th}} term is 8080
  2. 2The common difference dd of an arithmetic progression whose 1st1^{\text{st}} term is 33 and 5th5^{\text{th}} term is 3535
  3. 3The 4th4^{\text{th}} term of an arithmetic progression with first term 22 and common difference 33
  4. 4The sum of the first 33 terms of a geometric progression with first term 11 and common ratio 33

Cevap

The correct order from smallest to largest value is: the common ratio of the GP (4), the common difference of the AP (8), the 4th term of the AP (11), and the sum of the first 3 terms of the GP (13).
Evaluating each progression property gives numerical values of 4, 8, 11, and 13 respectively. Ordering these from least to greatest results in the order: common ratio of the GP (4), common difference of the AP (8), 4th term of the AP (11), and sum of the first 3 terms of the GP (13).

Adım Adım Çözüm

1
Calculate the value for the first item (common ratio rr)
Using T4=T2⋅r2T_4 = T_2 \cdot r^2, we have 80=5r2  ⟹  r2=16  ⟹  r=480 = 5 r^2 \implies r^2 = 16 \implies r = 4.
The terms of a GP follow Tn=arn−1T_n = a r^{n-1}.
2
Calculate the value for the second item (common difference dd)
Using T5=a+4dT_5 = a + 4d, we get 35=3+4d  ⟹  4d=32  ⟹  d=835 = 3 + 4d \implies 4d = 32 \implies d = 8.
The nth term of an AP is given by Tn=a+(n−1)dT_n = a + (n-1)d.
3
Calculate the value for the third item (4th4^{\text{th}} term of AP)
T4=2+(4−1)(3)=2+9=11T_4 = 2 + (4 - 1)(3) = 2 + 9 = 11.
Direct application of the AP nth term formula.
4
Calculate the value for the fourth item (Sum of first 33 terms of GP)
S3=1+3+9=13S_3 = 1 + 3 + 9 = 13 (or using Sn=a(rn−1)r−1=1(33−1)3−1=13S_n = \frac{a(r^n - 1)}{r - 1} = \frac{1(3^3 - 1)}{3 - 1} = 13).
Sum of a finite geometric sequence.
5
Compare and arrange the calculated numerical values in ascending order
4<8<11<134 < 8 < 11 < 13.
Ordering the numbers establishes the correct item sequence.

Anahtar Kavram

Evaluation of terms, differences, ratios, and sums in Arithmetic and Geometric Progressions.
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Konular

Arithmetic and Geometric Progressions (AP and GP)

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