Question

find the first term and the common difference of an arithmetic progression in which t5=3x-y and t11=5y

t5 refers to term 5 and t11 refers to term 11

Answer 1

first term: 5x-5y

common difference: -x/2 + y

Answer 2

The common difference is d = -0.5x + y and the first term is 5x-5y

Geometric progression

The nth term of the geometric progression is expressed as:

$T_n=a+(n-1)d$

where;

• a is the first term
• r is the common  ratio
• n is the number of terms

The fifth term of the sequence is expressed as:

$T_5=a+4d=3x-y$

$T_{11}=a+10d=3x-y$

Take the difference of the terms:

$4d-10d=3x-y-5y\\-6d=3x-6y\\d= -0.5x+y$

Get the first term:

a = 3x - y - 4(-0.5x+y)

a = 3x - y + 2x - 4y

a = 5x - 5y

Hence the common difference is d = -0.5x + y and the first term is 5x-5y

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