Arithmetic sequence

Determine the sum of the first 12 terms of an AP (arithmetic sequence) if a4 is equal to 7 and a8 is equal to minus 1.

Correct answer:

s =  24

Step-by-step explanation:


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Dr. Math
To find the sum of the first 12 terms of an arithmetic progression (AP), we are given:
- a4 = 7
- a8 = -1

Here’s the step-by-step solution:

Step 1:

Recall the Formula for the n -th Term of an AP
The n -th term of an AP is given by:
a_n = a1 + (n-1)d

where:
- a1 is the first term,
- d is the common difference,
- n is the term number.

Step 2:

Write Equations for a4 and a8
Using the formula for the n -th term:
1. For a4 :
 
  a4 = a1 + 3d = 7
 

2. For a8 :
 
  a8 = a1 + 7d = -1
 

Step 3:

Solve for a1 and d
Subtract the equation for a4 from the equation for a8 :
(a1 + 7d) - (a1 + 3d) = -1 - 7

4d = -8

d = -2


Now substitute d = -2 into the equation for a4 :
a1 + 3(-2) = 7

a1 - 6 = 7

a1 = 13

Step 4:

Find the Sum of the First 12 Terms
The sum of the first n terms of an AP is given by:
S_n = n/2 ( 2a1 + (n-1)d )

Substitute n = 12 , a1 = 13 , and d = -2 :
S12 = 12/2 ( 2(13) + (12-1)(-2) )

S12 = 6 ( 26 + 11(-2) )

S12 = 6 ( 26 - 22 )

S12 = 6 · 4

S12 = 24

Final Answer:


The sum of the first 12 terms of the AP is:

24







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