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(i) Find a formula for the sum S of any four consecutive odd numbers. (ii) Hence, find the value of S when the greatest odd number is -17 .

(i) Find a formula for the sum S S of any four consecutive odd numbers.\newline(ii) Hence, find the value of S S when the greatest odd number is −17-17 .

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Full solution

Q. (i) Find a formula for the sum S S of any four consecutive odd numbers.\newline(ii) Hence, find the value of S S when the greatest odd number is −17-17 .
  1. Define Odd Numbers: Let's denote the first odd number as nn. Since odd numbers are 22 units apart, the next three consecutive odd numbers would be n+2n+2, n+4n+4, and n+6n+6.
  2. Calculate Sum Formula: To find the formula for the sum SS of these four consecutive odd numbers, we add them together: S=n+(n+2)+(n+4)+(n+6)S = n + (n+2) + (n+4) + (n+6).
  3. Simplify Formula: Simplify the formula by combining like terms: S=4n+12S = 4n + 12.
  4. Find Value of n: Now, we need to find the value of SS when the greatest odd number is −17-17. Since the greatest number in our sequence is n+6n+6, we set n+6=−17n+6 = -17.
  5. Solve for n: Solve for n: n=−17−6n = -17 - 6.
  6. Calculate n: Calculate nn: n=−23n = -23.
  7. Substitute nn into Formula: Now that we have the value of nn, we can find SS by substituting nn into our formula S=4n+12S = 4n + 12.
  8. Calculate S: Calculate S: S=4(−23)+12S = 4(-23) + 12.
  9. Simplify Calculation: Simplify the calculation: S=−92+12S = -92 + 12.
  10. Final Calculation: Final calculation for SS: S=−80S = -80.

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