Question

the third term of an arithmetic progression is 4x - 2y and the 9th term is 10 x - 8 y find the common difference
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Answer 1

$\bold{\huge{\purple{\underline{ Solution }}}}$

Given :-

• The third term of an AP arithmetic progression is 4x - 2y
• The 9th term of an AP is 10x - 8y

To Find :-

• We have to find the common difference of the given AP?

Let's Begin :-

We know that,

For nth term of an AP

$\bold{\red{ an = a1 + (n - 1)d }}$

• Here, a1 is the first term of an AP
• n is the number of terms
• d is the common difference

We have ,

$\sf{ a3 = 4x - 2y ...eq(1)}$

$\sf{ a9 = 10x - 8y ...eq(2)}$

But, From above formula :-

$\sf{ a3 = a1 + (3 - 1)d}$

$\sf{ a3 = a1 + 2d...eq(3)}$

And

$\sf{ a9 = a1 + (9 - 1)d}$

$\sf{ a9 = a1 + 8d...eq(4)}$

From eq(1) and eq( 3) :-

$\sf{ a1 + 2d = 4x - 2y }$

$\sf{ a1 = 4x - 2y - 2d ...eq(5)}$

From eq(2) and eq( 4) :-

$\sf{ a1 + 8d = 10x - 8y }$

$\sf{ a1 = 10x - 8y - 8d ...eq(6)}$

From eq( 5) and eq(6) :-

$\sf{ 4x - 2y - 2d = 10x - 8y - 8d }$

$\sf{ -2d + 8d = 10x - 4x - 8y + 2y }$

$\sf{ 6d = 6x - 6y }$

$\sf{ 6d = 6(x - y) }$

$\sf{ d = }{\sf{\dfrac{ 6(x - y) }{6}}}$

$\bold{\blue{ d = x - y }}$

Hence, The common difference of the given AP is x - y.

Answer 2

The common difference of the given arithmetic progression is; d = x - y

What is the nth term of an arithmetic sequence?

Formula for the nth term of an arithmetic sequence is;

aₙ = a + (n - 1)d

where;

a is first term

n is position of term in the sequence

d is common difference

Thus;

a₃ = 4x - 2y

a₉ = 10x - 8y

Using the general formula, we know that;

a₃ = a + (3 - 1)d

a₃ = a + 2d   ----(eq 1)

a₉ = a + 8d   -----(eq 2)

Subtract eq 1 from eq 2 to get;

a₉ - a₃ = 6d

Put the given values of a₃ and a₉  to get;

10x - 8y - (4x - 2y) = 6d

6x - 6y = 6d

divide through by 6 to get;

d = x - y

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