The 4^{\text{th}} term of an arithmetic progression (AP) is 15 and the 9^{\text{th}} term … — JAMB UTME

The $4^{\text{th}}$ term of an arithmetic progression (AP) is $15$ and the $9^{\text{th}}$ term is $35$ . What is the sum of the first $12$ terms of the progression?

$300$ Cevap
B
$276$
C
$600$
D
$228$

Cevap

The sum of the first

12

terms of the progression is

300

Subtracting the

4^{\text{th}}

term from the

9^{\text{th}}

term gives

5d = 35 - 15 = 20

, so the common difference

d = 4

. Substituting

d = 4

into the

4^{\text{th}}

term expression

a + 3d = 15

yields

a = 3

. Using the sum formula

S_n = \frac{n}{2}[2a + (n-1)d]

for

n = 12

a = 3

, and

d = 4

gives

S_{12} = 6 \times (6 + 44) = 300

Adım Adım Çözüm

Set up the term equations using

T_n = a + (n-1)d

a + 3d = 15

and

a + 8d = 35

The

n^{\text{th}}

term formula for an AP is

T_n = a + (n-1)d

Solve the system of equations for

a

and

d

Subtracting the first equation from the second yields

5d = 20 \Rightarrow d = 4

. Substituting

d = 4

into

a + 3(4) = 15

gives

a = 3

Subtracting eliminates

a

to find the common difference

d

, which is then used to calculate the first term

a

Calculate the sum of the first

12

terms

S_{12}

S_{12} = \frac{12}{2}[2(3) + (12-1)4] = 6[6 + 44] = 6 \times 50 = 300

Applying the AP sum formula

S_n = \frac{n}{2}[2a + (n-1)d]

with

n = 12

a = 3

, and

d = 4

Anahtar Kavram

Arithmetic Progression

n^{\text{th}}

term and sum calculations

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru