expand (1+2x)^7 using binomial theorem
Answers
Step-by-step explanation:
Write out Pascal's triangle as far as the row which begins
1
,
6
...
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This gives you the sequence of coefficients for
(
a
+
b
)
6
:
1
,
6
,
15
,
20
,
15
,
6
,
1
Then we can account for the factor of
2
of the
2
x
term, by multiplying by a sequence of powers of
2
:
1
,
2
,
4
,
8
,
16
,
32
,
64
to get:
1
,
12
,
60
,
160
,
240
,
192
,
64
Hence:
(
1
+
2
x
)
6
=
1
+
12
x
+
60
x
2
+
160
x
3
+
240
x
4
+
192
x
5
+
64
x
6
Answer:
(1+2x)⁷ = 1+14x + 84x² + 280x³ + 560x⁴ + 672x⁵ + 448x⁶+ x128x⁷
Step-by-step explanation:
Required to expand (1+2x)⁷ using binomial theorem
Recall the formula
Binomial theorem:
(1+x)ⁿ = ⁿC₀+ⁿC₁(x) +ⁿC₂(x)² + ⁿC₃(x)³ + ................... ⁿCₙ(x)ⁿ ---------------(1)
Solution:
Here n = 7
ⁿC₀ = ⁷C₀ = 1
ⁿC₁ = ⁷C₁ = 7
ⁿC₂ = ⁷C₂ =
ⁿC₃ = ⁷C₃ =
ⁿC₄ = ⁷C₄ = ⁷C₃ = 35
ⁿC₅ = ⁷C₅ = ⁷C₂ =
ⁿC₆= ⁷C₆ = ⁷C₁ = 7
ⁿC₇ = ⁷C ₇ = 1
Substituting these values and the value of x as 2x we get in equation (1) we get
(1+2x)⁷= 1+7(2x) +21(2x)² + 35(2x)³ + 35(2x)⁴ + 21(2x)⁵ + 7(2x)⁶+ 1(2x)⁷
= 1+14x + 84x² + 280x³ + 560x⁴ + 672x⁵ + 448x⁶+ x128x⁷
(1+2x)⁷ = 1+14x + 84x² + 280x³ + 560x⁴ + 672x⁵ + 448x⁶+ x128x⁷
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