Answer:
For 1: The first term is 10 and the common difference is [tex]\frac{3}{2}[/tex]
For 2: The value of n is 27
Step-by-step explanation:
The n-th term of the progression is given as:
[tex]a_n=a_1+(n-1)d[/tex]
where,
[tex]a_1[/tex] is the first term, n is the number of terms and d is the common difference
The sum of n-th terms of the progression is given as:
[tex]S_n=\frac{n}{2}[2a_1+(n-1)d][/tex]
where,
[tex]S_n[/tex] is the sum of nth terms
The 11th term of the progression:
[tex]25=a_1+10d[/tex] .......(1)
Sum of first 4 numbers:
[tex]49=\frac{4}{2}[2a_1+3d[/tex] ......(2)
Forming equations:
[tex]98=8a_1+12d[/tex]
[tex]25=a_1+10d[/tex] ( × 8)
The equations become:
[tex]98=8a_1+12d[/tex]
[tex]200=8a_1+80d[/tex]
Solving above equations, we get:
[tex]102=68d\\\\d=\frac{102}{68}=\frac{3}{2}[/tex]
Putting value in equation (1):
[tex]25=a_1+10\frac{3}{2}\\\\a_1=[25-15]=10[/tex]
Hence, the first term is 10 and the common difference is [tex]\frac{3}{2}[/tex]
The nth term is given as:
[tex]49=10+(n-1)\frac{3}{2}[/tex]
Solving the above equation:
[tex]39=(n-1)\frac{3}{2}\\\\n-1=26\\\\n=27[/tex]
Hence, the value of n is 27
