Study the patterns and write down :
(a) Row 5
(b) Row n
(c) Row (n+1)

a) Row 5: 75 - 216 - 51

b) 3n² - (n+1)³ - (71-4n)

c) 3(n+1)² - (n+2)³ - (67-4n)

Step-by-step explanation:

Column 3:

Arithmetic sequence with common difference: 63-67 = -4

an = 67 + (n-1)(-4)

an = 67 - 4n + 4

an = 71 - 4n

Row 5: 3×5² - (5+1)³ - (71-(5×4))

Row 5: 75 - 216 - 51

Row n: 3n² - (n+1)³ - (71-4n)

Row n+1: 3(n+1)² - (n+1+1)³ - 71-4(n+1)

Row n+1: 3(n+1)² - (n+2)³ - (67-4n)

Column 1

Tn= 3n²

row n= 3n²

row n+1= 3(n+1)²

When n=5,

T₅ = 3(5)²

T₅ = 75

Thus, row 5= 75

Column 2

Tn= (n+1)³

row n= (n+1)³

row n+1

= (n+1+1)³

= (n+2)³

When n=5,

T₅ = (5+1)³

T₅ = 6³

T₅ = 216

row 5= 216

Column 3

Notice that the terms decrease by 4 with each row.

This means that there is a common difference of -4.

Common difference formula

Tn= a +d(n-1)

where a is the first term and d is the common difference.

Tn= 67 -4(n-1)

Now, simplify the formula

Tn= 67 -4n +4

Tn= 71 -4n

Row n= 71 -4n

Row n+1

= 71 -4(n+1)

= 71 -4n -4

= 67 -4n

when n=5,

T₅= 71 -4(5)

T₅= 51

Row 5= 51

Row n is the general formula for the pattern. To find the n+1 row, susbt. n+1 into n in the formula in row n.

answer for #1 is c

answer for #2 is d

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