Question

Find if 1000 is a term of the AP 25, 28, 31….. or not.

Answer 1

Hint: How would you imagine an AP term so far away from the first term that when you sit down to find every other term in the AP that till you reach the final term; you end up spending a significant portion of time? Simple. Just check for the common difference and compute it in such a way that some multiple of the common difference when added with the first term will give the required term.

Formulas used:
Nth term of AP is given by
${T_N} = {T_0} + nD$
Where ${T_N}$ is the Nth term, ${T_0}$ is the zeroth term, D is the common difference and n is the number of terms.

Complete step by step solution:
When working with this question, we have to form an equation for the same. We know that the Nth term of the equation is given by
${T_N} = {T_0} + nD$
Where ${T_N}$ is the Nth term, ${T_0}$ is the zeroth term, D is the common difference and n is the number of terms.
If we look at the progression, we find that the common difference is 3 which is equal to $28 - 25$ or $31 - 28$. First term here is 25.
Now, putting everything in the equation, we get
\[\
  {T_N} = {T_0} + nD \\
  1000 = 25 + n \times 3 \\
\ \]
For 1000 to be a term of the progression, n must be a whole number. So, we find n.
\[\
  1000 = 25 + n \times 3 \\
  1000 - 25 = 3n \\
  975 = 3n \\
  n = \dfrac{{975}}{3} \\
  n = 325 \\
\ \]

Which makes 1000 a part of this AP and the 375th term to be exact.

Note:
Here three points need to be kept in mind. Common difference is the difference between two consecutive terms only. Taking the difference of the first and the third term will not give a common difference but twice of it. Secondly, we take the first term as ${T_0}$ or term zero to simplify the equation and so the term ${T_N}$ will be the Nth term. And when written in term sequence it will be the $N + 1$th term if ${T_0}$ is the first term. Lastly n needed to be a whole number for 1000 to be a part of this AP. That was because you have first term, second term, third term, you never have 1.5th term or any decimal representation of the term.