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

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 ${{\text{T}}_{\text{n}}}{\text{ = a + (n - 1)d}}$

$${{\text{T}}_{\text{n}}}{\text{ = a + (n - 1)d}} \\\ \Rightarrow {\text{27 = a and}} \\\ \Rightarrow {\text{24 = a + d}} \\\$$

On solving both the above equation it is clear that

$${\text{a = 27, substituting its value in 24 = a + d, we get}} \\\ {\text{24 = 27 + d}} \\\ {\text{d = - 3}} \\\$$

Now , we have to calculate the number of terms up to which sums up to -30

$${{\text{S}}_{\text{n}}}{\text{ = }}\dfrac{{\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 = }}\dfrac{{\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}} \\\$$

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.