Need help on APGP H2 maths questions!
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1) A sheet of graph paper is marked in 1mm squares. A triangular region R is bounded by the x-axis, the line 500mm and the line y=(1/20)x. Find the number of complete 1mm squares which have all their verticies either inside R or on the boundary of R.
2)An arithmetic progression has n terms and a common difference d, where d>0. Prove that the difference between the sum of the last k terms nd the sum of the first k terms is (n-k)kd.
3)The fifth, tenth and twentieth terms of a convergent geometric progression, G, are the first three consecutive terms of arithmetic progression, A. Determine the common ratio of G.
4)A geometric series has first term 5 and common ratio 5^x.
i)Find the set of values of x for which the sum to infinity of the geometric series exists.
ii)Find the smallest n such that the sum of the first n terms exceeds 98% of the sum to infinity when x = -0.1.
Thanks for helping me out with my maths question! I really appreciate it!
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Hi,
Draw a diagram for Q1.
Q3, Q4 are basic questions that require you to apply properties of AP/GP.
For Q3, let a and r be T_1 and common ratio of G.
We are given that T_20 - T_10 = T_10 - T_5,
where T_5 = ar^4, T_10 = ar^9 and T_20 = ar^19.
For Q4, do recall the condition of common ratio for sum to infinity to exist.
I will address Q2 in detail only.
Let T_1 = a for the AP.
Sum of first k terms
= T_1 + ... + T_k
= k/2 ( T_1 + T_k )
= k/2 [ a + a + (k - 1)d ]
= k/2 [ 2a + (k - 1)d ] -- (1)
Sum of last k terms
= T_(n - k + 1) + ... + T_n
= k/2 ( T_(n - k + 1) + T_n )
= k/2 [ a + (n - k)d + a + (n - 1)d ]
= k/2 [ 2a + (2n - k - 1)d ] -- (2)
Difference = (2) - (1)
Thanks!
Cheers,
Wen Shih