Calculate the numerical values of the following sequence and progression quantities, then … — JAMB UTME

Calculate the numerical values of the following sequence and progression quantities, then arrange them in ascending order (from smallest to largest):

1The common ratio $r$ of a geometric progression (GP) with positive terms whose $2^{\text{nd}}$ term is $6$ and $5^{\text{th}}$ term is $162$ .
2The common difference $d$ of an arithmetic progression (AP) whose $3^{\text{rd}}$ term is $11$ and $7^{\text{th}}$ term is $27$ .
3The $5^{\text{th}}$ term of an arithmetic progression with first term $a = 2$ and common difference $d = 1.5$ .
4The sum to infinity $S_{\infty}$ of a geometric progression with first term $a = 5$ and common ratio $r = \frac{1}{2}$ .

Cevap

The correct ascending order of the numerical values is: the common ratio of the GP (value = 3), the common difference of the AP (value = 4), the 5th term of the AP (value = 8), and the sum to infinity of the GP (value = 10).

Evaluating each sequence property yields numerical values: the GP common ratio equals 3, the AP common difference equals 4, the AP 5th term equals 8, and the GP sum to infinity equals 10. Ordering these values from least to greatest produces the sequence 3, 4, 8, 10.

Adım Adım Çözüm

Calculate the value for the first quantity (AP common difference)

d = 4

From

a + 6d = 27

and

a + 2d = 11

, subtract to find

4d = 16 \implies d = 4

Calculate the value for the second quantity (GP common ratio)

r = 3

From

ar^4 = 162

and

ar = 6

, divide to obtain

r^3 = 27 \implies r = 3

Calculate the value for the third quantity (GP sum to infinity)

S_{\infty} = 10

Apply the sum to infinity formula

S_{\infty} = \frac{a}{1-r} = \frac{5}{1 - 0.5} = 10

Calculate the value for the fourth quantity (AP 5th term)

T_5 = 8

Apply the nth term formula

T_5 = a + 4d = 2 + 4(1.5) = 8

Sort the calculated values in ascending order

3 < 4 < 8 < 10

Comparing the values gives

3

(GP common ratio),

4

(AP common difference),

8

(AP 5th term), and

10

(GP sum to infinity).

Anahtar Kavram

Nth term, common difference, common ratio, and sum to infinity calculations for AP and GP sequences.

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