The sum of the first n terms of an arithmetic progression (AP) is given by S_n = 2n^2 + 3n… — JAMB UTME

The sum of the first $n$ terms of an arithmetic progression (AP) is given by $S_n = 2n^2 + 3n$ . The $3^{\text{rd}}$ term of this AP is equal to the $2^{\text{nd}}$ term of a geometric progression (GP). If the common ratio of the GP is $2$ , what is the sum of the first $4$ terms of the GP?

$97.5$ Cevap
B
$127.5$
C
$48.75$
D
$45.5$

Cevap

97.5

Evaluating

S_3 - S_2

gives the 3rd AP term as

27 - 14 = 13

. Setting the 2nd GP term

a(2) = 13

yields

a = 6.5

. The sum of the first 4 terms of the GP is

6.5 \times (2^4 - 1) = 6.5 \times 15 = 97.5

Adım Adım Çözüm

Find the 3rd term (

T_3

) of the AP using the given sum formula

S_n = 2n^2 + 3n

T_3 = S_3 - S_2 = [2(3)^2 + 3(3)] - [2(2)^2 + 3(2)] = [18 + 9] - [8 + 6] = 27 - 14 = 13

The

n

-th term of a sequence is equal to

S_n - S_{n-1}

Determine the first term (

a

) of the GP

Since

G_2 = 13

and common ratio

r = 2

a \cdot r^{2-1} = 13 \implies 2a = 13 \implies a = 6.5

The

n

-th term of a GP is given by

G_n = a r^{n-1}

Calculate the sum of the first 4 terms of the GP

S_4 = \frac{a(r^4 - 1)}{r - 1} = \frac{6.5(2^4 - 1)}{2 - 1} = 6.5 \times 15 = 97.5

The sum of the first

n

terms of a GP with

r > 1

S_n = \frac{a(r^n - 1)}{r - 1}

Anahtar Kavram

Combining Arithmetic Progression sum formula with Geometric Progression term and sum formulas

Tahmini Süre:2m 0s

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