Answer: a₂₁ = 65
Step-by-step explanation:
The Sum of an Arithmetic Progression is the sum of the first term plus the sum of the last term divided by 2 and multiplied by the number of terms.
[tex]a_1\ \text{is the first term}\\a_n=a_1+d(n-1)\quad \text{is the value of the nth term}\\\\[/tex]
Let's find the 16th term (n = 16)
[tex]a_{16}=a_1+d(16-1)\\\\.\quad =a_1+15d[/tex]
Now let's find the sum of the first 16 terms. This will be Equation 1:
[tex]S_{16}=\dfrac{(a_1)+(a_1+15d)}{2}\times 16=240\\\\\\.\qquad 8(2a_1+15d)=240\\\\\\.\qquad 2a_1+15d=30\qquad \leftarrow \text{Equation 1}[/tex]
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Repeat what we did above for the next 4 terms (n = 17 to n = 20). This will be Equation 2:
[tex]a_{17}=a_1+d(17-1)\\\\.\quad =a_1+16d\\\\\\a_{20}=a_1+d(20-1)\\\\.\quad =a_1+19d[/tex]
[tex]S_{17-20}=\dfrac{(a_1+16d)+(a_1+19d)}{2}\times 4=220\\\\\\.\qquad 2(2a_1+35d)=220\\\\\\.\qquad 2a_1+35d=110\qquad \leftarrow \text{Equation 2}[/tex]
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Now we have a system of equations. Solve using the Elimination Method:
2a₁ + 15d = 30 → -1(2a₁ + 15d = 30) → -2a₁ - 15d = -30
2a₁ + 35d = 110 → 1(2a₁ + 35d = 110) → 2a₁ + 35d = 110
20d = 80
d = 4
Input d = 4 into one the equations to solve for a₁:
Equation 1: 2a₁ + 15d = 30
2a₁ + 15(4) = 30
2a₁ + 60 = 30
2a₁ = -30
a₁ = -15
Given a₁ = -15 and d = 4, we can find the next term (n = 21)
[tex]a_n=a_1+d(n-1)\\\\a_{21}=-15+4(21-1)\\\\.\quad =-15+4(20)\\\\.\quad = -15+80\\\\.\quad = 65[/tex]
