Question

If 2x, x + 10, 3x + 2 are A. P., find the value of x.

Answer 1

Hint:One of the property of arithmetic progression is common difference is same and we can find common difference by subtracting \[{{1}^{st}}\] term from \[{{2}^{nd}}\] term or \[{{2}^{nd}}\] term from \[{{3}^{rd}}\] term and hence equating both the expression we get the value of x.

Complete step-by-step answer:
In the question we are given three quantities 2x, x + 10, 3x + 2 and are said that they are in arithmetic progression or A.P. We have to find the value of x.
At first we briefly understood what arithmetic progression is.
In mathematics an arithmetic progression (AP) or an arithmetic sequence of numbers such that the difference between the consecutive terms is constant. Here difference means the second minus first. For instance, the sequence 5, 7, 9, 11, 13, 15 ……. Is an arithmetic progression with a common difference of 2.
If the initial term of an arithmetic is considered as a and common difference in successive terms is d, then the \[{{n}^{th}}\] term of a sequence (\[{{T}_{n}}\]) is denoted by:
\[{{T}_{n}}=a+\left( n-1 \right)d\]
And in general, \[{{T}_{n}}={{T}_{m}}+\left( n-m \right)d\].
A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes called an arithmetic progression. The sum of finite arithmetic progression is called an arithmetic series.
In the series there is a property that the common difference remains the same means that the difference between third and second term is equal to difference between second and first term.
Here the \[{{1}^{st}}\] term is 2x, \[{{2}^{nd}}\] term is x + 10 and \[{{3}^{rd}}\] term is 3x + 2.
So, common difference is \[{{2}^{nd}}\] term - \[{{1}^{st}}\] term or ${(x + 10) – 2x}$ or $(10 - x).$
Common difference will also be \[{{3}^{rd}}\] term - \[{{2}^{nd}}\] term or ${(3x + 2) - (x + 10)}$ or ${3x + 2 – x - 10}$ or $(2x - 8)$.
As the common difference remains the same we can write,
10 – x = 2x – 8
Now subtracting 10 from both the sides so we get,
10 – x – 10 = 2x – 8 – 10
Or, - x = 2x – 18
Now subtracting 2x from both the sides so we get,
-x – 2x = 2x – 18 – 2x
Or, -3x = - 18
Hence, the value of x is 6.

Note: One can also do the sum by another method. As the given three terms are in AP so they can apply formula that twice the \[{{2}^{nd}}\] term is equal to the sum of \[{{1}^{st}}\] and \[{{3}^{rd}}\] term.