Question -
Answer -
(i) i49 +i68 + i89 + i110
Let us simplify weget,
i49 +i68 + i89 + i110 = i (48+ 1) + i68 + i(88 + 1) + i(108+ 2)
= (i4)12 ×i + (i4)17 + (i4)22 × i +(i4)27 × i2
= i + 1 + i – 1 [sincei4 = 1, i2 = – 1]
= 2i
∴ i49 +i68 + i89 + i110 = 2i
(ii) i30 +i80 + i120
Let us simplify weget,
i30 +i80 + i120 = i(28 + 2) + i80 +i120
= (i4)7 ×i2 + (i4)20 + (i4)30
= – 1 + 1 + 1 [since i4 =1, i2 = – 1]
= 1
∴ i30 +i80 + i120 = 1
(iii) i + i2 +i3 + i4
Let us simplify weget,
i + i2 +i3 + i4 = i + i2 + i2×i+ i4
= i – 1 + (– 1) × i +1 [since i4 = 1, i2 = – 1]
= i – 1 – i + 1
= 0
∴ i + i2 +i3 + i4 = 0
(iv) i5 + i10 +i15
Let us simplify weget,
i5 + i10 +i15 = i(4 + 1) + i(8 + 2) + i(12+ 3)
= (i4)1×i+ (i4)2×i2 + (i4)3×i3
= (i4)1×i+ (i4)2×i2 + (i4)3×i2×i
= 1×i + 1 × (– 1) + 1× (– 1)×i
= i – 1 – i
= – 1
∴ i5 +i10 + i15 = -1
(v) [i592 +i590 + i588 + i586 + i584]/ [i582 + i580 + i578 + i576 +i574]
Let us simplify weget,
[i592 +i590 + i588 + i586 + i584]/ [i582 + i580 + i578 + i576 +i574]
= [i10 (i582 +i580 + i578 + i576 + i574)/ (i582 + i580 + i578 + i576 +i574)]
= i10
= i8 i2
= (i4)2 i2
= (1)2 (-1) [since i4 = 1, i2 =-1]
= -1
∴ [i592 +i590 + i588 + i586 + i584]/ [i582 + i580 + i578 + i576 +i574] = -1
(vi) 1 + i2 +i4 + i6 + i8 + … + i20
Let us simplify weget,
1 + i2 +i4 + i6 + i8 + … + i20 =1 + (– 1) + 1 + (– 1) + 1 + … + 1
= 1
∴ 1 + i2 +i4 + i6 + i8 + … + i20 =1
(vii) (1 + i)6 +(1 – i)3
Let us simplify weget,
(1 + i)6 +(1 – i)3 = {(1 + i)2 }3 + (1 –i)2 (1 – i)
= {1 + i2 +2i}3 + (1 + i2 – 2i)(1 – i)
= {1 – 1 + 2i}3 +(1 – 1 – 2i)(1 – i)
= (2i)3 +(– 2i)(1 – i)
= 8i3 +(– 2i) + 2i2
= – 8i – 2i – 2 [sincei3 = – i, i2 = – 1]
= – 10 i – 2
= – 2(1 + 5i)
= – 2 – 10i
∴ (1 + i)6 +(1 – i)3 = – 2 – 10i