Question

Which term of the AP 20, 17, 14, ……… ; is the first negative term?

$
  {\text{A}}{\text{. }}{{\text{8}}^{{\text{th}}}} \\
  {\text{B}}{\text{. }}{{\text{6}}^{{\text{th}}}} \\
  {\text{C}}{\text{. }}{{\text{9}}^{{\text{th}}}} \\
  {\text{D}}{\text{. }}{{\text{7}}^{{\text{th}}}} \\
$

Answer 1

Hint – To find the first negative term, we suppose the last non-negative term of the series is definitely close to zero, ${{\text{a}}_{\text{n}}} \simeq 0$. We use the formula of ${{\text{n}}^{{\text{th}}}}$term in AP to find the answer.

Complete Step-by-Step solution:
Given: 20, 17, 14, ……… are in AP

Here, the first term a = 20
The common difference d = 17 -20 = 14 -17 = -3
d is the difference between any two consecutive terms in the Arithmetic progression.
n is the number of terms in the equation.

Let the first negative term of the series be${{\text{t}}_{\text{n}}}$, in order for it to be negative it has to be less than zero.

The ${{\text{n}}^{{\text{th}}}}$term of an AP is given by the formula, ${{\text{t}}_{\text{n}}} = {\text{a + }}\left( {{\text{n - 1}}} \right){\text{d}}$

In our AP,
${{\text{t}}_{\text{n}}}$< 0,
⟹${{\text{t}}_{\text{n}}} = {\text{a + }}\left( {{\text{n - 1}}} \right){\text{d}}$
$
   \Rightarrow {\text{a + }}\left( {{\text{n - 1}}} \right){\text{d < 0}} \\
   \Rightarrow {\text{20 + }}\left( {{\text{n - 1}}} \right) - 3{\text{ < 0}} \\
   \Rightarrow {\text{20 - 3n + 3 < 0}} \\
   \Rightarrow {\text{23 < 3n}} \\
   \Rightarrow {\text{n > }}\dfrac{{23}}{3} \\
   \Rightarrow {\text{n > 7}}{\text{.666}} \\
$

Therefore, the first negative term is the 8th term.
Hence, Option A is the right answer.

Note – The key in solving such types of problems is to observe the first negative term of the series is less than zero. Then we apply the formula of the ${{\text{n}}^{{\text{th}}}}$term of the equation. We solve this equation for n, we must remember we must round off n to its nearest whole number value.