Arithmetic Progressions: Difficult Problems with Solutions

Problem 1
Given the arithmetic progression [tex]\% a_1,a_2,a_3,...[/tex], for which [tex]a_1=1[/tex] and [tex]d=a_2-a_1=4[/tex], find [tex]a_8[/tex].
Problem 2
Find the difference of the arithmetic progression [tex]{a_n}[/tex] if [tex]a_1=5[/tex] and [tex]a_3=15[/tex]
Problem 3
If [tex]{a_n}[/tex] is an arithmetic progression, for which [tex]a_5=14[/tex] and [tex]a_1=2[/tex], find [tex]a_3[/tex].
Problem 4
Let [tex]{a_n}[/tex] be an arithemtic progression, for which [tex]a_2+a_3+a_4=54[/tex]. Find [tex]a_3[/tex].
Problem 5
Let [tex]{a_n}[/tex] be an arithmetic progression, for which [tex]a_1+a_2+a_3=102[/tex] and [tex]a_1=15[/tex]. Find [tex]a_{10}[/tex].
Problem 6
Let [tex]\{a_n\}[/tex] be an arithmetic progression. If [tex]a_3+a_8+a_{10}+a_{16}+a_{18}+a_{23}=126[/tex], find the sum of the first 25 members of [tex]\{a_n\}[/tex].
Problem 7
Find the difference d of an arithmetic progression [tex]{a_n}[/tex], for which [tex]a_1=9[/tex] and [tex]S_5=15[/tex]
Problem 8
Find the absolute value of the difference of the arithmetic progression [tex]{a_n}[/tex] if [tex]a_1+a_2=5[/tex] and [tex]a_1^2+a_2^2=13[/tex]
Problem 9 sent by Shravan Ananth
If 6 times the sixth term of an arithmetic progression is equal to 9 times the 9th term, find the 15th term.
Problem 10
How many hours would it take for a biker to travel 54 km if the first hour he traveled 15 km and each subsequent hour he traveled 1 km less than the hour before?

Problem 11
Find the sum of the first three elements of an arithmetic progression, for which [tex]a_1+a_5=22[/tex] and [tex]a_8-a_5=6[/tex]
Problem 12
If [tex]{a_n}[/tex] is an arithmetic progression, for which [tex]a_{10}=15[/tex] and [tex]a_5=5[/tex], find [tex]a_1[/tex].
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