(i) i49 + i68 + i89 + i110
Now let us simplify we get,
i49 + i68 + i89 + i110= i(48 + 1) + i68 + i(88 + 1) + i(108 + 2)
= (i4)12 × i + (i4)17 + (i4)11 × i + (i4)27 × i2
= i + 1 + i – 1 [since i4 = 1, i2 = – 1]
= 2i
∴ i49 + i68 + i89 + i110 = 2i
(ii) i30 + i80 + i120
Now let us simplify we get,
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
Now let us simplify we get,
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
Now let us simplify we get,
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]
Now let us simplify we get,
[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
Now let us simplify we get,
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
Now let us simplify we get,
(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 [since i3 = – i, i2 = – 1]
= – 10i – 2
= – 2(1 + 5i)
= – 2 – 10i
∴ (1 + i)6 + (1 – i)3 = – 2 – 10i
