UPSC NDA I 2021 Mathematics Solved Paper

Concept
Let us consider sequence a1,a2,a3...an is an A.P.
Common difference " d′′=a2−a1=a3−a2=...=an−an−1
n‌th ‌ term of the A.P. is given by an=a+(n−1)d
Sum of the first n terms =S=(n∕2)[2a+(n−1)×d]=(n∕2)(a+1)
Where, a= First term, d= Common difference, n= number of terms, an=n‌th ‌ term, 1= Last term
Calculation:
Given, a0=2‌‌...(1)
S5=‌

1

4

(S10−S5)
⇒4S5+S5=S10
⇒5S5=S10
⇒5×‌

5

2

[a0+a0+4d]=‌

10

2

[a0+a0+9d]
⇒5×[2a0+4d]=2×[2a0+9d]
⇒10a0+20d=4a0+18d
⇒a0=‌

−d

3

∴d=−3a0=−3×2=−6
S10=‌

10

2

[a0+a0+9d]=5[2a0+9d]=5[4−54],‌ as ‌a0=2,d=−6
⇒S10=5×(−50)=−250