How many terms are to be added to make the sum 52 in the series (-8) +

1.	How many terms are to be added to make the sum 52 in the series (-8) + (-6) + (-4) + ……………… ? A) 31 B) 13 C) 3 D) 12
Answer» 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. Correct option is B) 13

How many terms are to be added to make the sum 52 in the series (-8) + (-6) + (-4) + ……………… ? A) 31 B) 13 C) 3 D) 12

Answer»

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.

Correct option is B) 13