Sequence & Series III(Lowersixth Science Mathematics)

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Challenge on GP sums and sigma basics.

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Sequence & Series III(Lowersixth Science Mathematics)


Froggy Jumps Sequence & Series III(Lowersixth Science Mathematics)Online version Challenge on GP sums and sigma basics. by YAKILI LMS 1 What is the sum of the first n terms of a GP with first term a and ratio r (r ≠ 1)? a a(r^n−1)/(r−1) b a(1+r^n)/(1+r) c a(1−r^n)/(1−r) 2 What is the condition for convergence of the sum to infinity of a GP? a \|r\| > 1 b r ≠ 0 c \|r\| < 1 3 AM and GM relate by which inequality for positive numbers? a AM ≥ GM b AM ≤ GM c GM > AM 4 What principle is used to prove statements for all n in mathematics? a Mathematical induction b Contradiction c Direct proof 5 Sigma notation is used to denote what? a Product b Difference c Summation 6 Which symbol denotes a summation from i = 1 to n? a ∑i=1^n b Π from i=1 to n c Σ from i=1 to n 7 Sum to infinity of a GP with a and r equals what? a a/(1+r) b ar/(1−r) c a/(1−r) 8 What is the sum of a GP when r = −1/2 and a = 6 for n terms? a 6(1−(−1/2)^n)/(1+1/2) b 6(1−(−1/2)^n)/(1−1/2) c 6(1+(−1/2)^n)/(1+1/2) 9 If a = 3 and r = 2, sum of first n terms is? a 3(2^n−1)/(2−1) b 3(2^n−1)/1 c 3(1−2^n)/(1−2) 10 What is the general term of a GP? a a_n = a + nr b a_n = a/n * r c a_n = ar^(n−1) 11 What does Σi=1^n i represent? a Sum of integers from 1 to n b Sum of even numbers c Product of 1 to n 12 Which is a property of sigma notation? a Non-associative b No distributive rule c Linearity: Σ(a+b)=Σa+Σb 13 Which is true for AM-GM equality case? a All different b All numbers equal c Sum is zero 14 Sum to infinity exists only if r is? a \|r\| > 1 b r = 0 only c \|r\| < 1 15 What is the sum of the first n natural numbers? a n(n+1)/2 b n(n−1)/2 c n^2/2 16 Which series can be summed with induction proof? a Only finite series b Most arithmetic sequences c Only geometric series 17 What is the formula for sum of GP with a and r when r ≠ 1? a a(1+r^n)/(1+r) b a(1−r^n)/(1−r) c a(r^n−1)/(r−1) 18 What does Sigma notation with lower index mean? a Start term index b Difference index c End term 19 If a GP has r = 1, what is the sum of n terms? a a/n b an c a + nr 20 What is the sum of a GP with a=5, r=0.5, n=3? a 5(1−0.5^3)/1.5 b 5(1−0.5^3)/0.5 c 5(1−0.5^3)/1 21 Which is a correct sigma property? a Σ(a_i×c)=Σa_i×c^2 b Σ(ca_i)=cΣa_i c Σ(a_i+c)=Σa_i+c 22 AM-GM applies to which numbers? a Negative numbers only b Non-negative numbers c Always positive 23 What is an arithmetic mean of 4 and 10? a 6 b 8 c 7 24 What is the geometric mean of 4 and 9? a 6 b 3 c 12 25 What is the sum of first n terms of GP when r ≠ 1? a a(1−r^n)/(1−r) b a(1+r^n)/(1+r) c a(r^n−1)/(1−r) 26 Which involves induction base case? a Assuming P(n) is true b Proving P(1) is true c Proving P(n+1) is true without P(n) 27 Which is a convergent GP requirement? a \|r\|<1 b r≤0 and \|r\|≥1 c r>1 28 What is the sum to infinity for a GP with a=8 and r=1/3? a 8/(1−1/3)=24 b 8/(1+1/3)=6 c 8/(1−1/3)=12 29 Which notation means sum over k from 1 to n? a Π_{k=1}^n b ∫_{1}^{n} c Σ_{k=1}^n 30 What is the sum of integers 1 to 100? a 10000 b 5050 c 505 31 In sigma notation, what does the index i do? a Specifies terms summed over b Represents a coefficient c Indicates product 32 Which is not a GP property? a Terms increase by adding constant b Ratios constant c Each term multiplies by r 33 Sum of GP when a=2, r=−1/2, n=4 equals? a 2(1−(−1/2)^4)/(1+1/2) b 2(1−(−1/2)^4)/(1−(−1/2)) c 2(1+(−1/2)^4)/(1−(−1/2)) 34 Which identity is used in sigma linearity? a Σ(a−b)=Σa−Σb^2 b Σ(a×b)=Σa×Σb c Σ(a+b)=Σa+Σb 35 Which sequence has an constant ratio? a Harmonic sequence b Geometric sequence c Arithmetic sequence 36 If S_n is sum of GP terms, S_n is function of? a a, r, n b Only n c Only a 37 What is the AM of numbers 2, 8, 18? a 6 b 8 c 9 38 Induction proof typically has base case and? a Hypothesis only b Inductive step c Counterexample 39 Sum of first n terms of AP is? a n/2(a+nd) b n(a+nd)/2 c n/2[2a+(n−1)d] 40 What does Σi^n i^2 represent? a Sum of squares 1^2+...+n^2 b Sum of cubes c Product of squares 41 Which is a correct GP term formula? a a_n = ar^(n−1) b a_n = a/n c a_n = a + nr^2 42 What is the property Σ(a_i b_i) if a_i constant? a Cannot simplify b Always zero c b_i can be pulled out 43 What is the sum of negatives in a GP with r<0? a Always positive b Depends on n and r c Always negative 44 What is the sum of 1+1/2+1/4+... to n terms? (a=1, r=1/2) a 1(1−(1/2)^n)/(1−1/2) b (1−(1/2)^n)/(1+1/2) c n/2 45 Which is not a valid sigma notation form? a Σ_{i=1}^n i b Σ_i=1^n i c ∑_{i=1}^n i 46 Which describes convergence of infinite GP? a Limit of terms tends to 0 when \|r\|<1 b Sum always finite c Only if a=0 47 What is the base of induction? a P(1) true b P(n) false c P(n+1) assumed 48 Which is a correct AM-GM example? a (x−y)/2 ≤ √(xy) b For x,y>0, (x+y)/2 ≥ √(xy) c x+y ≤ 2√(xy) 49 What is the sum of 3 terms of GP: a=4, r=3? a 4(1−27)/(1−3) b 4(27−1)/(1−3) c 4(1−27)/(3−1) 50 What does the symbol Σ denote? a Multiplication b Differentiation c Summation

Sequence & Series III(Lowersixth Science Mathematics)

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Sequence & Series III(Lowersixth Science Mathematics)Online version

by YAKILI LMS

What is the sum of the first n terms of a GP with first term a and ratio r (r ≠ 1)?

What is the condition for convergence of the sum to infinity of a GP?

AM and GM relate by which inequality for positive numbers?

What principle is used to prove statements for all n in mathematics?

Sigma notation is used to denote what?

Which symbol denotes a summation from i = 1 to n?

Sum to infinity of a GP with a and r equals what?

What is the sum of a GP when r = −1/2 and a = 6 for n terms?

If a = 3 and r = 2, sum of first n terms is?

What is the general term of a GP?

What does Σi=1^n i represent?

Which is a property of sigma notation?

Which is true for AM-GM equality case?

Sum to infinity exists only if r is?

What is the sum of the first n natural numbers?

Which series can be summed with induction proof?

What is the formula for sum of GP with a and r when r ≠ 1?

What does Sigma notation with lower index mean?

If a GP has r = 1, what is the sum of n terms?

What is the sum of a GP with a=5, r=0.5, n=3?

Which is a correct sigma property?

AM-GM applies to which numbers?

What is an arithmetic mean of 4 and 10?

What is the geometric mean of 4 and 9?

What is the sum of first n terms of GP when r ≠ 1?

Which involves induction base case?

Which is a convergent GP requirement?

What is the sum to infinity for a GP with a=8 and r=1/3?

Which notation means sum over k from 1 to n?

What is the sum of integers 1 to 100?

In sigma notation, what does the index i do?

Which is not a GP property?

Sum of GP when a=2, r=−1/2, n=4 equals?

Which identity is used in sigma linearity?

Which sequence has an constant ratio?

If S_n is sum of GP terms, S_n is function of?

What is the AM of numbers 2, 8, 18?

Induction proof typically has base case and?

Sum of first n terms of AP is?

What does Σi^n i^2 represent?

Which is a correct GP term formula?

What is the property Σ(a_i b_i) if a_i constant?

What is the sum of negatives in a GP with r<0?

What is the sum of 1+1/2+1/4+... to n terms? (a=1, r=1/2)

Which is not a valid sigma notation form?

Which describes convergence of infinite GP?

What is the base of induction?

Which is a correct AM-GM example?

What is the sum of 3 terms of GP: a=4, r=3?

What does the symbol Σ denote?