Question

Find the number of terms in the A.P 6,10,14,…,174.

Answer 1

Hint: Find the common difference of the A.P by subtracting the second term from the first term. Assume that there are n terms in the A.P. Use the fact that the ${{n}^{th}}$ term of an A.P is given by ${{a}_{n}}=a+\left( n-1 \right)d$. Substitute ${{a}_{n}}=174$ and the value of a and d and hence form an equation in n. Solve for n and hence find the number of terms of the A.P. Verify your answer.

Complete step-by-step answer:
Before solving the question, we need to understand the properties of A.P.
An A.P is a sequence of numbers in which there is increment/decrement per term and that increment/decrement is constant through the series, i.e. if a,b,c are the first three terms of an A.P then b-a = c-b. This constant increment/decrement is called the common difference of the A.P. The first term of the A.P is denoted by “a” and the common difference of the A.P is denoted by “d”. Hence, we have d = b-a.
Let there are n terms in the A.P
We have $a=6$ and $d={{a}_{2}}-{{a}_{1}}=10-6=4$
Also, we have ${{a}_{n}}=174$
We know that in an A.P if “a” is the first term, “d” is the common difference, then then ${{n}^{th}}$ term of the A.P is given by ${{a}_{n}}=a+\left( n-1 \right)d$
Substituting the values of $a,d,{{a}_{n}}$, we get
$174=6+\left( n-1 \right)4$
Subtracting 6 from both sides, we get
$\left( n-1 \right)4=168$
Dividing by 4 on both sides, we get
$n-1=42$
Adding 1 on both sides, we get
$n=43$
Hence there are 43 terms in the A.P.

Note: [1] Verification:
We can verify the correctness of our solution by checking that the ${{43}^{rd}}$ term of the A.P is 174
We have a = 6 and d = 4
Hence, we have
${{a}_{43}}=a+42d=6+42\times 4=174$
Hence the ${{43}^{rd}}$ term of the A.P is 174. Hence our solution is verified to be correct.