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Question
The first three terms of a geometric sequence are \(\ln {x^{16}}\), \(\ln {x^8}\), \(\ln {x^4}\), for \(x > 0\).
Find the common ratio.
Solve \(\sum\limits_{k = 1}^\infty {{2^{5 – k}}\ln x = 64} \).
Answer/Explanation
Markscheme
correct use \(\log {x^n} = n\log x\) A1
eg\(\,\,\,\,\,\)\(16\ln x\)
valid approach to find \(r\) (M1)
eg\(\,\,\,\,\,\)\(\frac{{{u_{n + 1}}}}{{{u_n}}},{\text{ }}\frac{{\ln {x^8}}}{{\ln {x^{16}}}},{\text{ }}\frac{{4\ln x}}{{8\ln x}},{\text{ }}\ln {x^4} = \ln {x^{16}} \times {r^2}\)
\(r = \frac{1}{2}\) A1 N2
[3 marks]
recognizing a sum (finite or infinite) (M1)
eg\(\,\,\,\,\,\)\({2^4}\ln x + {2^3}\ln x,{\text{ }}\frac{a}{{1 – r}},{\text{ }}{S_\infty },{\text{ }}16\ln x + \ldots \)
valid approach (seen anywhere) (M1)
eg\(\,\,\,\,\,\)recognizing GP is the same as part (a), using their \(r\) value from part (a), \(r = \frac{1}{2}\)
correct substitution into infinite sum (only if \(\left| r \right|\) is a constant and less than 1) A1
eg\(\,\,\,\,\,\)\(\frac{{{2^4}\ln x}}{{1 – \frac{1}{2}}},{\text{ }}\frac{{\ln {x^{16}}}}{{\frac{1}{2}}},{\text{ }}32\ln x\)
correct working (A1)
eg\(\,\,\,\,\,\)\(\ln x = 2\)
\(x = {{\text{e}}^2}\) A1 N3
[5 marks]
Question
An arithmetic sequence has \({u_1} = {\text{lo}}{{\text{g}}_c}\left( p \right)\) and \({u_2} = {\text{lo}}{{\text{g}}_c}\left( {pq} \right)\), where \(c > 1\) and \(p,\,\,q > 0\).
Show that \(d = {\text{lo}}{{\text{g}}_c}\left( q \right)\).
Let \(p = {c^2}\) and \(q = {c^3}\). Find the value of \(\sum\limits_{n = 1}^{20} {{u_n}} \).
Answer/Explanation
Markscheme
valid approach involving addition or subtraction M1
eg \({u_2} = {\text{lo}}{{\text{g}}_c}\,p + d,\,\,{u_1} – {u_2}\)
correct application of log law A1
eg \({\text{lo}}{{\text{g}}_c}\left( {pq} \right) = {\text{lo}}{{\text{g}}_c}\,p + {\text{lo}}{{\text{g}}_c}\,q,\,\,{\text{lo}}{{\text{g}}_c}\left( {\frac{{pq}}{p}} \right)\)
\(d = {\text{lo}}{{\text{g}}_c}\,q\) AG N0
[2 marks]
METHOD 1 (finding \({u_1}\) and d)
recognizing \(\sum { = {S_{20}}} \) (seen anywhere) (A1)
attempt to find \({u_1}\) or d using \({\text{lo}}{{\text{g}}_c}\,{c^k} = k\) (M1)
eg \({\text{lo}}{{\text{g}}_c}\,c\), \({\text{3}}\,{\text{lo}}{{\text{g}}_c}\,c\), correct value of \({u_1}\) or d
\({u_1}\) = 2, d = 3 (seen anywhere) (A1)(A1)
correct working (A1)
eg \({S_{20}} = \frac{{20}}{2}\left( {2 \times 2 + 19 \times 3} \right),\,\,{S_{20}} = \frac{{20}}{2}\left( {2 + 59} \right),\,\,10\left( {61} \right)\)
\(\sum\limits_{n = 1}^{20} {{u_n}} \) = 610 A1 N2
METHOD 2 (expressing S in terms of c)
recognizing \(\sum { = {S_{20}}} \) (seen anywhere) (A1)
correct expression for S in terms of c (A1)
eg \(10\left( {2\,{\text{lo}}{{\text{g}}_c}\,{c^2} + 19\,{\text{lo}}{{\text{g}}_c}\,{c^3}} \right)\)
\({\text{lo}}{{\text{g}}_c}\,{c^2} = 2,\,\,\,{\text{lo}}{{\text{g}}_c}\,{c^3} = 3\) (seen anywhere) (A1)(A1)
correct working (A1)
eg \({S_{20}} = \frac{{20}}{2}\left( {2 \times 2 + 19 \times 3} \right),\,\,{S_{20}} = \frac{{20}}{2}\left( {2 + 59} \right),\,\,10\left( {61} \right)\)
\(\sum\limits_{n = 1}^{20} {{u_n}} \) = 610 A1 N2
METHOD 3 (expressing S in terms of c)
recognizing \(\sum { = {S_{20}}} \) (seen anywhere) (A1)
correct expression for S in terms of c (A1)
eg \(10\left( {2\,{\text{lo}}{{\text{g}}_c}\,{c^2} + 19\,{\text{lo}}{{\text{g}}_c}\,{c^3}} \right)\)
correct application of log law (A1)
eg \(2\,{\text{lo}}{{\text{g}}_c}\,{c^2} = \,\,{\text{lo}}{{\text{g}}_c}\,{c^4},\,\,19\,{\text{lo}}{{\text{g}}_c}\,{c^3} = \,\,{\text{lo}}{{\text{g}}_c}\,{c^{57}},\,\,10\,\left( {{\text{lo}}{{\text{g}}_c}\,{{\left( {{c^2}} \right)}^2} + \,\,{\text{lo}}{{\text{g}}_c}\,{{\left( {{c^3}} \right)}^{19}}} \right),\,\,10\,\left( {{\text{lo}}{{\text{g}}_c}\,{c^4} + \,{\text{lo}}{{\text{g}}_c}\,{c^{57}}} \right),\,\,10\left( {{\text{lo}}{{\text{g}}_c}\,{c^{61}}} \right)\)
correct application of definition of log (A1)
eg \({\text{lo}}{{\text{g}}_c}\,{c^{61}} = 61,\,\,{\text{lo}}{{\text{g}}_c}\,{c^4} = 4,\,\,{\text{lo}}{{\text{g}}_c}\,{c^{57}} = 57\)
correct working (A1)
eg \({S_{20}} = \frac{{20}}{2}\left( {4 + 57} \right),\,\,10\left( {61} \right)\)
\(\sum\limits_{n = 1}^{20} {{u_n}} \) = 610 A1 N2
[6 marks]