Question

Write the correct number in the given boxes from the following A.P.
2, _ , 26.

Answer 1

Hint:Use the formula ${{a}_{n}}={{a}_{1}}+\left( n-1 \right)d$ and find out the common difference using the third term. Then use the common difference to calculate the second term by using the same formula.

Complete step by step answer:
As we have to find the missing number and it is given that the numbers are in A.P. therefore we will assume the terms as,
2, _ , 26.
Therefore assume first term to be ${{a}_{1}}=2$, second term be ‘${{a}_{2}}$’ and third term be, ${{a}_{3}}=26$ ……………… (1)
To proceed further in the solution we should know the formula of nth term of and A.P. given below,
Formula:
${{a}_{n}}={{a}_{1}}+\left( n-1 \right)d$
By using above formula we can write the second term as,
${{a}_{2}}={{a}_{1}}+\left( 2-1 \right)d$
$\therefore {{a}_{2}}={{a}_{1}}+d$ ………………………………. (2)
Also the third term be,
${{a}_{3}}={{a}_{1}}+\left( 3-1 \right)d$
${{a}_{3}}={{a}_{1}}+2d$
But as we know that the third term in the given A.P. is 26 and if we put this value and the values of equation (1) in the above equation we will get,
$\therefore 26=2+2d$
Taking 2 common from the above equation we will get,
$\therefore 26=2\times \left( 1+d \right)$
If we shift 2 on the left hand side of the equation we will get,
$\therefore \dfrac{26}{2}=\left( 1+d \right)$
By rearranging the above equation we will get,
$\therefore \left( 1+d \right)=\dfrac{26}{2}$
If we simplify the above equation we will get,
$\therefore 1+d=13$
$\therefore d=13-1$
$\therefore d=12$ ………………………………… (2)
Now, if we put the values of equation (1) and equation (3) in equation (2) we will get,
\[\therefore {{a}_{2}}=2+12\]
If we add 12 and 2 in the above equation we will get,
\[\therefore {{a}_{2}}=14\]
Therefore the correct number for the given A.P. is 14 and therefore the A.P. becomes, 2, 14, 26.

Note: You can also solve this problem by assuming the second term as ‘x’ and using the concept that the common difference between two terms of an A.P. is always the same. From that concept you will get the equation ‘x – 2 = 26 – x’ and using this equation you can easily find the solution in no time.