Correct option is (B) 13
Given sequence is -8, -6, -4, .......
\(\because\) \(a_2-a_1=-6-(-8)=2\)
\(a_3-a_2=-4-(-6)=2\)
\(\because\) \(a_3-a_2\) = \(a_2-a_1\)
\(\therefore\) Sequence -8, -6, -4, ....... is an A.P.
whose common difference is d = 2 & first term is \(a=a_1=-8.\)
Let n terms are to be added to make the sum 52 in the given series.
i.e., \(S_n=52\)
\(\Rightarrow\frac n2[2a+(n-1)d]=52\)
\(\Rightarrow\frac n2[2\times-8+(n-1)2]=52\) \((\because a=-8\;\&\;d=2)\)
\(\Rightarrow n[-16+2n-2]=52\times2=104\)
\(\Rightarrow2n^2-18n-104=0\)
\(\Rightarrow n^2-9n-52=0\)
\(\Rightarrow n^2-13n+4n-52=0\)
\(\Rightarrow n(n-13)+4(n-13)=0\)
\(\Rightarrow(n-13)(n+4)=0\)
\(\Rightarrow n-13=0\;or\;n+4=0\)
\(\Rightarrow n=13\;or\;n=-4\)
\(\because\) Number of terms never be negative.
\(\therefore n\neq-4\)
Therefore n = 13
Hence, 13 terms are to be added to make the sum 52 in the given series.
