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Word problems on Geometric Progression

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Question 41

Question:

Answer:

Question 42

Question: One side of a square is 8 cm. The midpoints of its sides are joined to form another square, whose midpoints are again joined to form one more square and this process is continued indefinitely. Find the sum of areas of all the squares formed.

Answer:
First square is ABCD. Second square is A₁ B₁ C₁ D₁. Third square is
A₂ B₂ C₂ D₂.
Let each side of square be a units.
Then the length of side of the second square is

The side if the third square is

Area of first square = a²

If the length of each side of the square is 8 cm, then the area is
2a² = 2(8)² = 128 sq cm.

Question 43

Question: The sum of the first n term of a certain series show that the terms of the series form a GP and find the first term and the common ratio.

Answer:

Question 44

Question: Find the sum of the squares of first n odd natural numbers.

Answer: S= 1² + 3² + 5²+.... to n terms

Question 45

Question: Sum the series

Answer: Let t_n = n^t^h term of the given series

(Note that sum of first n odd natural numbers is n²)

Question 46

Question: The sequence N of natural numbers is divided into groups as follows:

Show that the sum of the numbers in the n^t^h row is n(2n²+1).

Answer: We observe that the number of terms in 1^s^t row = 2, 2ⁿ^d row = 4, 3^r^d row=6 and so on. The number of terms in n^t^h row = 2n.
\ The number of numbers upto n^t^h row (inclusive of the n^t^h row).

If S_n= sum of (1+2+3+...n), then the required sum = S_n₍_n₊₁₎ - S_n₍_n_-₁₎