Example 16 A person has 2 parents, 4 grandparents, 8 great grandparents, and so on Find the number of his ancestors during the ten generations preceding his own.
Answers
Answer:
2046
Step-by-step explanation:
is said that a person has 2 parents, 4 grandparents, 8 great grandparents etc. So we can write this as a series.
Thus the number of ancestors can be written as 2,4,8,16……
This forms a G.P. which is a geometric progression which is a sequence in which each term is derived by multiplying or dividing the preceding term by a fixed number called the common ratio. It is denoted by r.
∴
Common ratio, r=Term 2Term 1=42=2
r=Term 3Term 2=84=2
Thus we got the common ratio, r = 2 and first term, a = 2.
It is said that we need to find the number of ancestors during 10 generations preceding his own, i.e. we need to find S10
.
n=10
, which is the number of terms.
Sum of n terms of a G.P. is given by the formula,
Sn=a(rn−1)r−1
.
Put n = 10, a = 2, r = 2 and find S10
.
∴S10=2(210−1)2−1∴S10=2(210−1)=2.210−2=210+1−2=211−2.∴S10=2048−2=2046.
∴
The value of 211=2048
, which is 2 multiplied 11times.
Thus we got the number of ancestors preceding the person as 2046.
Answer:
1024 is the answer of your question