(i) How many terms of the sequence 18,16,14,... should be taken so that their sum is zero?
(ii) How many terms are there in the A.P. whose first and fifth terms are −14 and 2 respectively and the sum of the terms is 40?
(iii) How many terms of the A.P. 9,17,25,... must be taken so that their sum is 636?
(iv) How many terms of the A.P. 63,60,57,... must be taken so that their sum is 693?
(v) How many terms of the A.P. 27,24,21... should be taken so that their sum is zero?
(i) Given AP: 18,16,14,........
Here, a=18,d=16−18=−2
We know that the Sum of nth term of the sequence is given by
Sn=n2[2a+(n−1)d]
Now, Sn=0
⇒n2[2a+(n−1)d]=0
⇒n2[2×18+(n−1)×−2]=0
⇒n2[36−2n+2]=0
⇒n2[38−2n]=0
⇒2n2[19−n]=0
⇒n1[19−n]=0
⇒n=0or,19−n=0
⇒n=19 [∵n≠0]
Hence, the sum of 19 terms of the given AP will be zero.
(ii) Given: First term, =a1=a=−14, Fifth term a5=2 and Sn=40
We know that the nth of an AP is given by an=a+(n−1)d
∴a5=a+(5−1)d=2
⇒−14+4×d=2
⇒4d=16
⇒d=4
Now, Sn=40
⇒n2[2a+(n−1)d]=40
⇒n2[2×−14+(n−1)4]=40
⇒n2[−28+4n−4]=40
⇒4n2[−7+n−1]=40
⇒2n1[−8+n]=40
⇒n[−8+n]=20
⇒n2−8n−20=0
⇒n2−10n+2n−20=0
⇒n(n−10)+2(n−10)=0
⇒(n+2)(n−10)=0
⇒(n+2)=0or,(n−10)=0
⇒n=−2or,n=10
∴ there are 10 terms in the given AP. (number of terms cannot be negative)
(iii) Given AP: 9,17,25,........
Here, a=9, and d=17−9=8
We know that the Sum of nth term of the sequence is given by
Sn=n2[2a+(n−1)d]
Now, Sn=636
⇒n2[2a+(n−1)d]=636
⇒n2[2×9+(n−1)×8]=636
⇒n2[18+8n−8]=636
⇒2n2[5+4n]=636
⇒n(5+4n)=636
⇒4n2+5n−636=0
⇒4n2−48n+53n+636=0
⇒4n(n−12)+53(n−12)=0
⇒(4n+53)(n−12)=0
⇒(4n+53)=0or,(n−12)=0
∴n=12 (∵n≠ negative )
Hence, the sum of 12 terms will be 636 of the given AP.
(iv) Given AP: 63,60,57,...
Sn=693
n2[2a+(n−1)d]=693
⇒n2[2×63+(n−1)(−3)]=693
⇒n[126−3n+3]=693×2
⇒3n2−129n+693×2=0
⇒n2−43n+462=0
⇒n2−22n−21n+462=0
⇒n(n−22)−21(n−22)=0
⇒(n−21)(n−22)=0
∴n=21or,n=22
Hence, the sum of 21 or 22 terms will be 693.
Given AP: 27,24,21...
Sn=0
n2[2a+(n−1)d]=0
⇒n2[2×27+(n−1)(−3)]=0
⇒n[54−3n+3]=0
⇒n[57−3n]=0
⇒3n[19−n]=0
⇒n[19−n]=0
⇒n=0or,[19−n]=0
⇒n=19 [∵n≠0]
Hence, the sum of 19 terms will be 0.