Question

The first two terms of an A.P are $$\text{27}$$ and $$\text{24}$$ respectively. How many terms of the progression are to be added to get -30?
A.15
B.20
C.25
D.18

Answer 1

Hint: Term of an A.P is denoted as $${\text{T}}_{\text{n}}\text{ = a + (n - 1)d}$$ , so we substitute the given value of $$\text{27}$$and $$\text{24}$$ in the equation and calculate the value of a and d then finally calculate the value of n for $${\text{S}}_{\text{n}}\text{ = - 30}$$.

Complete step-by-step answer:
As first two term of A.P are 27 and 24 so, equate it to $${\text{T}}_{\text{n}}\text{ = a + (n - 1)d}$$
$$\begin{gathered} {\text{T}}_{\text{n}}\text{ = a + (n - 1)d} \\ \Rightarrow \text{27 = a and} \\ \Rightarrow \text{24 = a + d} \end{gathered}$$
On solving both the above equation it is clear that
$$\begin{gathered} \text{a = 27, substituting its value in 24 = a + d, we get} \\ \text{24 = 27 + d} \\ \text{d = - 3} \end{gathered}$$
Now , we have to calculate the number of terms up to which sums up to -30
 $$\begin{gathered} {\text{S}}_{\text{n}}\text{ = }{\displaystyle \frac{\text{n}}{\text{2}}}\text{(2a + (n - 1)d)} \\ \text{On substituting the value of a, d and }{\text{S}}_{\text{n}}\text{,we get,} \\ \Rightarrow \text{ - 30 = }{\displaystyle \frac{\text{n}}{\text{2}}}\text{(2(27) + (n - 1)( - 3))} \\ \Rightarrow \text{ - 60 = n(57 - 3n)} \\ \text{On simplification we get,} \\ \Rightarrow \text{3}{\text{n}}^{\text{2}}\text{ - 57n - 60 = 0} \\ \text{On factorisation we get,} \\ \Rightarrow \text{3}{\text{n}}^{\text{2}}\text{ - 60n + 3n - 60 = 0} \\ \Rightarrow \text{3n(n - 20) + 3(n - 20) = 0} \\ \Rightarrow \text{(3n + 3)(n - 20) = 0} \\ \Rightarrow \text{n = 20 or n = - 1} \end{gathered}$$
As value of n cannot be negative,
Hence option (b) is the correct answer.

Note: An arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant. For example, the sequence 1, 2, 3, 4, ... is an arithmetic progression with common difference 1
Use the data given in the question carefully, place them and form the equation and solve it so that the correct value of required can be obtained.